do we really understand quantum mechanics strange correlations

Do we really understand quantum mechanics? Strange correlations,paradoxes, and theorems

F. Laloea)

Laboratoire de Physique de l’ENS, LKB, 24 rue Lhomond, F-75005 Paris, France

~Received 10 November 1998; accepted 29 January 2001!

This article presents a general discussion of several aspects of our present understanding of quantummechanics. The emphasis is put on the very special correlations that this theory makes possible:They are forbidden by very general arguments based on realism and local causality. In fact, thesecorrelations are completely impossible in any circumstance, except for very special situationsdesigned by physicists especially to observe these purely quantum effects. Another general pointthat is emphasized is the necessity for the theory to predict the emergence of a single result in asingle realization of an experiment. For this purpose, orthodox quantum mechanics introduces aspecial postulate: the reduction of the state vector, which comes in addition to the Schro¨dingerevolution postulate. Nevertheless, the presence in parallel of two evolution processes of the sameobject~the state vector! may be a potential source for conflicts; various attitudes that are possible toavoid this problem are discussed in this text. After a brief historical introduction, recalling how thevery special status of the state vector has emerged in quantum mechanics, various conceptualdifficulties are introduced and discussed. The Einstein–Podolsky–Rosen~EPR! theorem ispresented with the help of a botanical parable, in a way that emphasizes how deeply the EPRreasoning is rooted into what is often called ‘‘scientific method.’’ In another section theGreenberger–Horne–Zeilinger argument, the Hardy impossibilities, as well as the Bell–Kochen–Specker theorem are introduced in simple terms. The final two sections attempt to give a summaryof the present situation: One section discusses nonlocality and entanglement as we see it presently,with brief mention of recent experiments; the last section contains a~nonexhaustive! list of variousattitudes that are found among physicists, and that are helpful to alleviate the conceptual difficultiesof quantum mechanics. ©2001 American Association of Physics Teachers.

@DOI: 10.1119/1.1356698#

6

CONTENTS

I. HISTORICAL PERSPECTIVE. .. . . . . . . . . . . . . . . 656A. Three periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 657

1. Prehistory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6572. The undulatory period. . . . . . . . . . . . . . . . . . 6573. Emergence of the Copenhagen

interpretation. . . . . . . . . . . . . . . . . . . . . . . . . 658B. The status of the state vector. . . . . . . . . . . . . . . . 658

1. Two extremes and the orthodoxsolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658

2. An illustration. . . . . . . . . . . . . . . . . . . . . . . . . 659II. DIFFICULTIES, PARADOXES. . . . . . . . . . . . . . . . 659

A. Von Neumann’s infinite regress. . . . . . . . . . . . . 660B. Wigner’s friend. . . . . . . . . . . . . . . . . . . . . . . . . . . 661C. Schrodinger’s cat. . . . . . . . . . . . . . . . . . . . . . . . . 661D. Unconvincing arguments. . . . . . . . . . . . . . . . . . . 662

III. EINSTEIN, PODOLSKY, AND ROSEN. . . . . . . . 662A. A theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662B. Of peas, pods, and genes. . . . . . . . . . . . . . . . . . . 663

1. Simple experiments; no conclusion yet. . . . . 6632. Correlations; causes unveiled. . . . . . . . . . . . 6633. Transposition to physics. . . . . . . . . . . . . . . . . 664

IV. QUANTITATIVE THEOREMS: BELL,GREENBERGER–HORNE–ZEILINGER,HARDY, BELL–KOCHEN–SPECKER. . . . . . . . 666

A. Bell inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . 6661. Two spins in a quantum singlet state. . . . . . 6662. Proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

655 Am. J. Phys.69 ~6!, June 2001 http://ojps.aip.org/a

3. Contradiction with quantum mechanicsand with experiments. . . . . . . . . . . . . . . . . . . 667

4. Generality of the theorem. . . . . . . . . . . . . . . 667B. Hardy’s impossibilities. . . . . . . . . . . . . . . . . . . . . 668C. GHZ equality. . . . . . . . . . . . . . . . . . . . . . . . . . . . 669D. Bell–Kochen–Specker; contextuality. . . . . . . . . 671

V. NONLOCALITY AND ENTANGLEMENT:WHERE ARE WE NOW?. . . . . . . .. . . . . . . . . . . . 672A. Loopholes, conspiracies.. . . . . . . . . . . . . . . . . . . 672B. Locality, contrafactuality. . . . . . . . . . . . . . . . . . . 674C. ‘‘All-or-nothing coherent states;’’

decoherence. .. . . . . . . . . . . . . . . . . . . . . . . . . . . 6751. Definition and properties of the states. . . . . 6752. Decoherence. . . . . . . . . . . . . . . . . . . . . . . . . . 676

D. Quantum cryptography, teleportation. . . . . . . . . 6771. Sharing cryptographic keys by quantum

measurements. . . . . . . . . . . . . . . . . . . . . . . . . 6772. Teleporting a quantum state. . . . . . . . . . . . . 678

E. Quantum computing and information.. . . . . . . . 679VI. VARIOUS INTERPRETATIONS. . . . . . . . . . . . . . 680

A. Common ground; ‘‘correlationinterpretation’’.. . . . . . . . . . . . . . . . . . . . . . . . . . 680

B. Additional variables. . . . . . . . . . . . . . . . . . . . . . . 6821. General framework. . . . . . . . . . . . . . . . . . . . . 6822. Bohmian trajectories. . . . . . . . . . . . . . . . . . . 683

C. Modified ~nonlinear! Schrodinger dynamics. . . . 6841. Various forms of the theory. . . . . . . . . . . . . . 6842. Physical predictions. . . . . . . . . . . . . . . . . . . . 685

655jp/ © 2001 American Association of Physics Teachers

gh

opten

onetby

tahyss

ratis

-esfe

asb

eenit

ctob

llyrnI

ec

ute

tehin,thiseaaro

ss

adys on

dis-whave, is a20

ionset-hrif

then-cyoodhe

anict

ble

the

inentassonicseven

er-the

im-heef.

Ref.dis-n-attentsed

er-

cs

of

is-

ewizeshe

r-

rceery

in-

D. History interpretation. . . . . . . . . . . . . . . . . . . . . . 6861. Histories, families of histories. . . . . . . . . . . . 6862. Consistent families. . . . . . . . . . . . . . . . . . . . . 6873. Quantum evolution of an isolated system. . . 6874. Comparison with other interpretations. . . . . 6885. A profusion of points of view; discussion. . . 689

E. Everett interpretation. . . . . . . . . . . . . . . . . . . . . . 690VII. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

Quantum mechanics describes physical systems throumathematical object, the state vectoruC&, which replaces po-sitions and velocities of classical mechanics. This is an enmous change, not only mathematically, but also conceally. The relations betweenuC& and physical properties armuch less direct than in classical mechanics; the distabetween the formalism and the experimental predictileaves much more room for discussions about the interprtion of the theory. Actually, many difficulties encounteredthose who tried~or are still trying! to ‘‘really understand’’quantum mechanics are related to questions pertaining toexact status ofuC&: For instance, does it describe the physicreality itself, or only some partial knowledge that we mighave of this reality? Does it fully describe ensemble of stems only ~statistical description!, or one single system awell ~single events!? Assume that, indeed,uC& is affected byan imperfect knowledge of the system; is it then not natuto expect that a better description should exist, at leasprinciple? If so, what would be this deeper and more precdescription of the reality?

Another confusing feature ofuC& is that, for systems extended in space~for instance, a system made of two particlat very different locations!, it gives an overall description oall its physical properties in a single block from which thnotion of space seems to have disappeared; in some cthe physical properties of the two remote particles seem tocompletely ‘‘entangled’’~the word was introduced by Schro¨-dinger in the early days of quantum mechanics! in a waywhere the usual notions of space–time and local events sto become dimmed. Of course, one could think that thistanglement is just an innocent feature of the formalism wno special consequence: For instance, in classical elemagnetism, it is often convenient to introduce a choicegauge for describing the fields in an intermediate step,we know very well that gauge invariance is actually fupreserved at the end. But, and as we will see below, it tuout that the situation is different in quantum mechanics:fact, a mathematical entanglement inuC& can indeed haveimportant physical consequences on the result of expments, and even lead to predictions that are, in a sense,tradictory with locality~we will see below in what sense!.

Without any doubt, the state vector is a rather curioobject to describe reality; one purpose of this article isdescribe some situations in which its use in quantum mchanics leads to predictions that are particularly unexpecAs an introduction, and in order to set the stage for tdiscussion, we will start with a brief historical introductiowhich will remind us of the successive steps from whichpresent status ofuC& emerged. Paying attention to historynot inappropriate in a field where the same recurrent idare so often rediscovered; they appear again and agsometimes almost identical over the years, sometimesmodeled or rephrased with new words, but in fact moreless unchanged. Therefore, a look at the past is not neceily a waste of time!

656 Am. J. Phys., Vol. 69, No. 6, June 2001

a

r-u-

cesa-

helt-

line

es,e

em-

hro-fut

sn

ri-on-

so-d.s

e

sin,e-rar-

I. HISTORICAL PERSPECTIVE

The founding fathers of quantum mechanics had alreperceived the essence of many aspects of the discussionquantum mechanics; today, after almost a century, thecussions are still lively and, if some very interesting neaspects have emerged, at a deeper level the questionsnot changed so much. What is more recent, neverthelessgeneral change of attitude among physicists: Until aboutyears ago, probably as a result of the famous discussbetween Bohr, Einstein, Schro¨dinger, Heisenberg, Pauli, dBroglie, and others~in particular at the famous Solvay meeings, Ref. 1!, most physicists seemed to consider that ‘‘Bowas right and proved his opponents to be wrong,’’ eventhis was expressed with more nuance. In other words,majority of physicists thought that the so-called ‘‘Copehagen interpretation’’ had clearly emerged from the infanof quantum mechanics as the only sensible attitude for gscientists. As we all know, this interpretation introduced tidea that modern physics must contain indeterminacy asessential ingredient: It is fundamentally impossible to predthe outcome of single microscopical events; it is impossito go beyond the formalism of the wave function~or itsequivalent, the state vector1 uC&! and complete it; for somephysicists, the Copenhagen interpretation also includesdifficult notion of ‘‘complementarity’’... even if it is truethat, depending on the context, complementarity comesmany varieties and has been interpreted in many differways! By and large, the impression of the vast majority wthat Bohr had eventually won the debate with Einstein,that discussing again the foundations of quantum mechaafter these giants was pretentious, useless, and maybebad taste.

Nowadays, the attitude of physicists is much more modate concerning these matters, probably partly becausecommunity has better realized the nonrelevance of the ‘‘possibility theorems’’ put forward by the defenders of tCopenhagen orthodoxy, in particular by Von Neumann, R2 ~see also Refs. 3–5, as well as the discussion given in6!; another reason is, of course, the great impact of thecoveries and ideas of J. Bell, Ref. 7. At the turn of the cetury, it is probably fair to say that we are no longer sure ththe Copenhagen interpretation is the only possible consisattitude for physicists—see for instance the doubts expresin Ref. 8. Alternative points of view are considered as pfectly consistent: theories including additional variables~or‘‘hidden variables’’!,2 see Refs. 9 and 10; modified dynamiof the state vector, Refs. 4, 11, 12, and 13~nonlinear and/orstochastic evolution!; at the other extreme we have pointsview such as the so-called ‘‘many worlds interpretation’’~ormultibranched universe interpretation!, Ref. 14, or more re-cently other interpretations such as that of ‘‘decoherent htories,’’ Ref. 15 ~the list is nonexhaustive!. All these inter-pretations will be discussed in Sec. VI. For a recent revicontaining many references, see Ref. 16, which emphasadditional variables, but which is also characteristic of tvariety of positions among contemporary scientists,3 as wellas an older but very interesting debate published inPhysicsToday ~Ref. 17!; another very useful source of older refeences is the 1971 AJP ‘‘Resource Letter’’~Ref. 18!. Butrecognizing this variety of positions should not be the souof misunderstandings! It should also be emphasized vclearly that, until now, no new fact whatsoever~or no newreasoning! has appeared that has made the Copenhagenterpretation obsolete in any sense.

656F. Laloe

hpthestmre

tu. 2tue

inde

iche

ungtten

ort

talaoneraDep

o-ceroc-w

oncedereohetu

ckthb-caicthouicin

anp

ichr

theavehath aan

calowwe

th

-rialandv-notve,im

deallyandthat

andw

hys-re-e; iniontlytro-iale

, butall

y

la-of

hro

tedtheesallne

bence,ervenyofmainris

; in

A. Three periods

Three successive periods may be distinguished in thetory of the elaboration of the fundamental quantum concethey have resulted in the point of view that we may call ‘‘torthodox interpretation,’’ with all provisos that have jubeen made above. Here we give only a brief historical sumary, but we refer the reader who would like to know moabout the history of the conceptual development of quanmechanics to the book of Jammer, Ref. 19; see also Reffor detailed discussions of fundamental problems in quanmechanics, one could also look for references such as R21, 22, and 8 or those given in Ref. 18.

1. Prehistory

Planck’s name is obviously the first that comes to mwhen one thinks about the birth of quantum mechanics: Hthe one who introduced the famous constanth, which nowbears his name, even if his method was phenomenologHis motivation was actually to explain the properties of tradiation in thermal equilibrium~blackbody radiation! by in-troducing the notion of finite grains of energy in the calclation of the entropy, later interpreted by him as resultifrom discontinuous exchange between radiation and maIt is Einstein who, later, took the idea more seriously areally introduced the notion of quantum of light~whichwould be named ‘‘photon’’ much later!, in order to explainthe wavelength dependence of the photoelectric effect—fgeneral discussion of the many contributions of Einsteinquantum theory, see Ref. 23.

One should nevertheless realize that the most imporand urgent question at the time was not so much to expfine details of the properties of radiation–matter interactior the peculiarities of the blackbody radiation; it was, rathto understand the origin of the stability of atoms, that is ofmatter which surrounds us and of which we are made!spite several attempts, explaining why atoms do not collaalmost instantaneously was still a complete challengephysics. One had to wait a little bit more, until Bohr intrduced his celebrated atomic model, to see the appearanthe first elements allowing treatment of the question. He pposed the notion of ‘‘quantized permitted orbits’’ for eletrons, as well as that of ‘‘quantum jumps’’ to describe hothey would go from one orbit to another, during radiatiemission processes for instance. To be fair, we must conthat these notions have now almost disappeared from mophysics, at least in their initial forms; quantum jumps areplaced by a much more precise theory of spontaneemission in quantum electrodynamics. But, on the othand, one may also see a resurgence of the old quanjumps in the modern use of the postulate of the wave pareduction. After Bohr came Heisenberg, who introducedtheory that is now known as ‘‘matrix mechanics,’’ an astract intellectual construction with a strong philosophicomponent, sometimes close to positivism; the classphysical quantities are replaced by ‘‘observables,’’ maematically matrices, defined by suitable postulates withmuch help of the intuition. Nevertheless, matrix mechancontained many elements which turned out to be buildblocks of modern quantum mechanics!

In retrospect, one can be struck by the very abstractsomewhat mysterious character of atomic theory at thisriod of history; why should electrons obey such rules whforbid them to leave a given class of orbits, as if they we


is-s;

-

m0;mfs.

is

al.

-

r.d

ao

ntin,,

ll-

sein

of-

dern

usrm

ete

lal-t

sg

de-

e

miraculously guided on simple trajectories? What wasorigin of these quantum jumps, which were supposed to hno duration at all, so that it would make no sense to ask wwere the intermediate states of the electrons during sucjump? Why should matrices appear in physics in suchabstract way, with no apparent relation with the classidescription of the motion of a particle? One can guess hrelieved many physicists felt when another point of vieemerged, a point of view which looked at the same timmuch simpler and in the tradition of the physics of the 19century: the undulatory~or wave! theory.

2. The undulatory period

It is well known that de Broglie was the first who introduced the idea of associating a wave with every mateparticle; this was soon proven to be correct by DavissonGermer in their famous electron diffraction experiment. Neertheless, for some reason, at that time de Broglie didproceed much further in the mathematical study of this waso that only part of the veil of mystery was raised by h~see for instance the discussion in Ref. 24!. It is sometimessaid that Debye was the first who, after hearing aboutBroglie’s ideas, remarked that in physics a wave generhas a wave equation: The next step would then be to trypropose an equation for this new wave. The story addsthe remark was made in the presence of Schro¨dinger, whosoon started to work on this program; he successfullyrapidly completed it by proposing the equation which nobears his name, one of the most basic equations of all pics. Amusingly, Debye himself does not seem to havemembered the event. The anecdote may not be accuratfact, different reports about the discovery of this equathave been given and we will probably never know exacwhat happened. What remains clear anyway is that the induction of the Schro¨dinger equation is one of the essentmilestones in the history of physics. Initially, it allowed onto understand the energy spectrum of the hydrogen atomwe now know that it also gives successful predictions forother atoms, molecules and ions, solids~the theory of bandsfor instance!, etc. It is presently the major basic tool of manbranches of modern physics and chemistry.

Conceptually, at the time of its introduction, the undutory theory was welcomed as an enormous simplificationthe new mechanics; this is particularly true because Sc¨-dinger and others~Dirac, Heisenberg! promptly showed howit allowed one to recover the predictions of the complicamatrix mechanics from more intuitive considerations onproperties of the newly introduced ‘‘wave function’’—thsolution of the Schro¨dinger equation. The natural hope wathen to be able to extend this success, and to simplifyproblems raised by the mechanics of atomic particles: Owould replace it by a mechanics of waves, which wouldanalogous to electromagnetic or sound waves. For instaSchrodinger thought initially that all particles in the universlooked to us like point particles just because we obsethem at a scale which is too large; in fact, they are ti‘‘wave packets’’ which remain localized in small regionsspace. He had even shown that these wave packets resmall ~they do not spread in space! when the system undestudy is a harmonic oscillator... . Alas, we now know that this only one of the very few special cases where this is truegeneral, they do constantly spread in space!

657F. Laloe

thulea

dri-toisll

thkn-nm

tinio

eict,

et aris

ticse,othactioasurthcangr t

hade

ic

no

own

e

n

e,acnd

et

ntalentgleveandl’’er-

eialics.

othse

alw,

theintbythed tolyingpt

buthe

tionistalicalersfor-bil-

; isket

d: Ifem,theenectsac--are-ex-theeer

tetes

theg

ad-

3. Emergence of the Copenhagen interpretation

It did not take a long time before it became clear thatundulatory theory of matter also suffers from very seriodifficulties—actually so serious that physicists were soonto abandon it. A first example of difficulty is provided bycollision between particles, where the Schro¨dinger wavespreads in all directions, exactly as the water wave stirrea pond by a stone thrown into it; but, in all collision expements, particles are observed to follow well-defined trajecries which remain perfectly localized, going in some precdirection. For instance, every photograph taken in the cosion chamber of a particle accelerator shows very clearlyparticles never get ‘‘diluted’’ in all space! This remarstimulated the introduction, by Born, of the probabilistic iterpretation of the wave function. Another difficulty, evemore serious, arises as soon as one considers systemsof more than one single particle: Then, the Schro¨dinger waveis no longer an ordinary wave since, instead of propagain normal space, it propagates in the so-called ‘‘configuratspace’’ of the system, a space which has 3N dimensions fora system made ofN particles! For instance, already for thsimplest of all atoms, the hydrogen atom, the wave whpropagates in six dimensions~if spins are taken into accounfour such waves propagate in six dimensions!; for a macro-scopic collection of atoms, the dimension quickly becoman astronomical number. Clearly the new wave was not asimilar to classical waves, which propagate in ordinaspace; this deep difference will be a sort of Leitmotiv in thtext,4 reappearing under various aspects here and there.5

In passing, and as a side remark, it is amusing to nothat the recent observation of the phenomenon of BoEinstein condensation in dilute gases~Ref. 25! can be seenin a sense, as a sort of realization of the initial hopeSchrodinger: This condensation provides a case wheremany-particle matter wave does propagate in ordinary spBefore condensation takes place, we have the usual situaThe atoms belong to a degenerate quantum gas, which hbe described by wave functions defined in a huge configtion space. But, when they are completely condensed,are restricted to a much simpler many-particle state thatbe described by the same wave function, exactly as a siparticle. In other words, the matter wave becomes similaa classical field with two components~the real part and theimaginary part of the wave function!, resembling an ordinarysound wave for instance. This illustrates why, somewparadoxically, the ‘‘exciting new states of matter’’ provideby Bose–Einstein condensates are not an example of antreme quantum situation; they are actually more classthan the gases from which they originate~in terms of quan-tum description, interparticle correlations, etc.!. Conceptu-ally, of course, this remains a very special case and doessolve the general problem associated with a naive viewthe Schro¨dinger waves as real waves.

The purely undulatory description of particles has ndisappeared from modern quantum mechanics. In additioBorn and Bohr, Heisenberg~Ref. 26!, Jordan, Dirac~Ref.27!, and others played an essential role in the appearanca new formulation of quantum mechanics~Ref. 20!, whereprobabilistic and undulatory notions are incorporated insingle complex logical edifice. The now classical Copehagen interpretation of quantum mechanics~often also called‘‘orthodox interpretation’’! incorporates both a progressivdeterministic evolution of the wave function/state vectorcording to the Schro¨dinger equation, as well as a seco


esd

in

-ei-at

ade

gn

h

sll

y

e–

fee.n:to

a-eynleo

t

x-al

otf

to

of

a-

-

postulate of evolution that is often called the ‘‘wave packreduction’’ ~or also ‘‘wave function collapse’’!. The Schro¨-dinger equation in itself does not select precise experimeresults, but keeps all of them as potentialities in a coherway; forcing the emergence of a single result in a sinexperiment is precisely the role of the postulate of the wapacket reduction. In this scheme, separate postulatesequations are therefore introduced, one for the ‘‘naturaevolution of the system, another for measurements pformed on it.

B. The status of the state vector

With two kinds of evolution, it is no surprise if the statvector should get, in orthodox quantum theory, a nontrivstatus—actually it has no equivalent in all the rest of phys

1. Two extremes and the orthodox solution

Two opposite mistakes should be avoided, since b‘‘miss the target’’ on different sides. The first is to endorthe initial hopes of Schro¨dinger and to decide that the~many-dimension! wave function directly describes the physicproperties of the system. In such a purely undulatory viethe position and velocities of particles are replaced byamplitude of a complex wave, and the very notion of poparticle becomes diluted; but the difficulties introducedthis view are now so well known—see the discussion inpreceding section—that few physicists seem to be temptesupport it. Now, by contrast, it is surprising to hear relativeoften colleagues falling to the other extreme, and endorsthe point of view where the wave function does not attemto describe the physical properties of the system itself,just the information that we have on it—in other words, twave function should get a relative~or contextual! status,and become analogous to a classical probability distribuin usual probability theory. Of course, at first sight, thwould bring a really elementary solution to all fundamenproblems of quantum mechanics: We all know that classprobabilities undergo sudden jumps, and nobody considthis as a special problem. For instance, as soon as new inmation becomes available to us on any system, the probaity distribution that we associate with it changes suddenlythis not the obvious way to explain the sudden wave pacreduction?

One first problem with this point of view is that it woulnaturally lead to a relative character of the wave functiontwo observers had different information on the same systshould they use different wave functions to describesame system?6 In classical probability theory, there would bno problem at all with ‘‘observer-dependent’’ distributioprobabilities, but standard quantum mechanics clearly rejthis possibility: It certainly does not attribute such a charter to the wave function.7 Moreover, when in ordinary probability theory a distribution undergoes a sudden ‘‘jump’’ tomore precise distribution, the reason is simply that more pcise values of the variables already exist—they actuallyisted before the jump. In other words, the very fact thatprobability distribution reflected our imperfect knowledgimplies the possibility for a more precise description, closto the reality of the system itself. But this is in compleopposition with orthodox quantum mechanics, which negathe very idea of a better description of the reality thanwave function. In fact, introducing the notion of pre-existinvalues is precisely the basis of unorthodox theories withditional variables~hidden variables!! So the advocates of this

658F. Laloe

i-c

io

llveunn

mav

e

tio

msyi

‘‘thtiheenatevetioei

f

aomer

no

stin

p

thThhie-t i

litwro

lyrnhevth

ay.cals

f theme-avediffi-om-or,ote

c-ty:andil-ndit.

bywith-wen-

ngr islare-eralysi-,nc-lly

isite,avelist

efule-c-e ac-

t

tum

tionuss

ckethein

in-theec-theallemrmalnt

‘‘information interpretation’’8 are often advocates of addtional variables ~often called hidden variables—see SeVI B and note 2!, without being aware of it! It is thereforeimportant to keep in mind that, in the classical interpretatof quantum mechanics, the wave function~or state vector!givestheultimate physical description of the system, with aits physical properties; it is neither contextual, nor obserdependent; if it gives probabilistic predictions on the resof future measurements, it nevertheless remains inherecompletely different from an ordinary classical distributioof probabilities.

If none of these extremes is correct, how should we cobine them? To what extent should we consider that the wfunction describes a physical system itself~realistic interpre-tation!, or rather that it contains only the information that wmay have on it~positivistic interpretation!, presumably insome sense that is more subtle than a classical distribufunction? This is not an easy question, and various authanswer the question with different nuances; we will coback to this question in Sec. II B, in particular in the discusion of the ‘‘Schro¨dinger cat paradox.’’ Even if it not so easto be sure about what the perfectly orthodox interpretationwe could probably express it by quoting Peres in Ref. 29:state vector is not a property of a physical system, but rarepresents an experimental procedure for preparing or tesone or more physical systems;’’ we could then add anotquotation from the same article, as a general comm‘‘quantum theory is incompatible with the proposition thmeasurements are processes by which we discover somknown and preexisting property.’’ In this context, a wafunction is an absolute representation, but of a preparaprocedure rather than of the isolated physical system itsnevertheless, since this procedure may also imply someformation on the system itself~for instance, in the case orepeated measurements of the same physical quantity!, wehave a sort of intermediate situation where none of theswers above is completely correct, but where they are cbined in a way that emphasizes the role of the whole expmental setup.

2. An illustration

Just as an illustration of the fact that the debate isclosed, we take a quotation from a recent article~Ref. 30!which, even if taken out of its context, provides an intereing illustration of the variety of nuances that can exist withthe Copenhagen interpretation~from the context, it seemsclear that the authors adhere to this interpretation!; after criti-cizing erroneous claims of colleagues concerning the prouse of quantum concepts, they write: ‘‘~One! is led astray byregarding state reductions as physical processes, ratheraccepting that they are nothing but mental processes.’’authors do not expand much more on this sentence, wthey relate on a ‘‘minimalistic interpretation of quantum mchanics;’’ actually they even give a general warning that idangerous to go beyond it~‘‘Van Kampen’s caveat’’!. Nev-ertheless, let us try to be bold and to cross this dangerousfor a minute; what is the situation then? We then see thatdifferent attitudes become possible, depending on the perties that we attribute to the Schro¨dinger evolution itself: Isit also a ‘‘purely mental process,’’ or is it of a completedifferent nature and associated more closely with an extereality? Implicitly, the authors of Ref. 30 seem to favor tsecond possibility—otherwise, they would probably hamade a more general statement about all evolutions of


.

n

rlttly

-e

onrse-

s,aerngrt:

un-

nlf;n-

n--

i-

t

-

er

ane

ch

s

neop-

al

ee

state vector—but let us examine both possibilities anywIn the first case, the relation of the wave function to physireality is completely lost and we meet all the difficultiementioned in the preceding paragraph as well as some onext section; we have to accept the idea that quantumchanics has nothing to say about reality through the wfunction ~if the word reality even refers to any well-definenotion!!. In the second case, we meet the conceptual dculties related to the coexistence of two processes of cpletely different nature for the evolution of the state vectas discussed in the next section. What is interesting is to nthat Peres’s point of view~at the end of the preceding setion!, while also orthodox, corresponds to neither possibiliIt never refers to mental process, but just to preparationtests on physical systems, which is clearly different; thislustrates the flexibility of the Copenhagen interpretation athe variety of ways that different physicists use to describe

Another illustration of the possible nuances is provideda recent note published by the same author togetherFuchs~Ref. 31! entitled, ‘‘Quantum theory needs no ‘interpretation.’ ’’ These authors explicitly take a point of viewhere the wave function is not absolute, but observer depdent: ‘‘it is only a mathematical expression for evaluatiprobabilities and depends on the knowledge of whoevedoing the computing.’’ The wave function becomes simito a classical probability distribution which, obviously, dpends on the knowledge of the experimenter, so that sevdifferent distributions can be associated with the same phcal system~if there are several observers!. On the other handas mentioned above, associating several different wave futions with one single system is not part of what is usuacalled the orthodox interpretation~except, of course, for atrivial phase factor!.

To summarize, the orthodox status of the wave functionindeed a subtle mixture between different, if not opposconcepts concerning reality and the knowledge that we hof this reality. Bohr is generally considered more as a reathan a positivist or an operationalist~Ref. 19!; he wouldprobably have said that the wave function is indeed a ustool, but that the concept of reality cannot properly be dfined at a microscopic level only; it has to include all maroscopic measurement apparatuses that are used to havcess to microscopic information~we come back to this poinin more detail in Sec. III B 3!. In this context, it is under-standable why he once even stated that ‘‘there is no quanconcept’’ ~Ref. 32!!

II. DIFFICULTIES, PARADOXES

We have seen that, in most cases, the wave funcevolves gently, in a perfectly predictable and continuoway, according to the Schro¨dinger equation; in some caseonly ~as soon as a measurment is performed!, unpredictablechanges take place, according to the postulate of wave pareduction. Obviously, having two different postulates for tevolution of the same mathematical object is unusualphysics; the notion was a complete novelty when it wastroduced, and still remains unique in physics, as well assource of difficulties. Why are two separate postulates nessary? Where exactly does the range of application offirst stop in favor of the second? More precisely, amongthe interactions—or perturbations—that a physical systcan undergo, which ones should be considered as no~Schrodinger evolution!, which ones are a measureme

659F. Laloe

haps

ysthcht

theaueueta

tosa

of

ecslio

l

eaht—i

th’’e

eann

si

o

o

ut

allyord

um-tilleras

hetheithible

em-

-la-in

reent

ffi-ro-cro-do

1/2the

with

theence-tor,sedrent

ver,ul-

thenof

thepicint-

s on

-p-forthe

on?dic-

venen-o

aph.nensntlyour-

ress.

~wave packet reduction!? Logically, we are faced with aproblem that did not exist before, when nobody thought tmeasurements should be treated as special processes inics. We learn from Bohr that we should not try to transpoour experience of the everyday world to microscopic stems; this is fine, but where exactly is the limit betweentwo worlds? Is it sufficient to reply that there is so muroom between macroscopic and microscopic sizes thatexact position of the border does not matter?9

Moreover, can we accept that, in modern physics,‘‘observer’’ should play such a central role, giving to ththeory an unexpected anthropocentric foundation, as intronomy in the middle ages? Should we really refuse asscientific to consider isolated~unobserved! systems, becauswe are not observing them? These questions are difficalmost philosophical, and we will not attempt to answthem here. Rather, we will give a few characteristic quotions, which illustrate10 various positions.

~i! Bohr ~second Ref. 19, page 204!: ‘‘There is no quan-tum world... it is wrong to think that the task of physics isfind out how Nature is. Physics concerns what we canabout Nature.’’

~ii ! Heisenberg~same reference, page 205!: ‘‘But the at-oms or the elementary particles are not real; they formworld of potentialities or possibilities rather than onethings and facts.’’11

~iii ! Jordan~as quoted by Bell in Ref. 33!: ‘‘observationsnot only disturb what has to be measured, theyproduceit. Ina measurement of position, the electron is forced to a dsion. We compel it to assume a definite position; previouit was neither here nor there, it had not yet made its decisfor a definite position... .’’

~iv! Mermin ~Ref. 6!, summarizing the ‘‘fundamentaquantum doctrine’’~orthodox interpretation!: ‘‘the outcomeof a measurement is brought into being by the act of msurement itself, a joint manifestation of the state of tprobed system and the probing apparatus. Precisely howparticular result of an individual measurement is obtainedHeisenberg’s transition from the possible to the actual—inherently unknowable.’’

~v! Bell ~Ref. 34!, speaking of ‘‘modern’’ quantum theory~Copenhagen interpretation!: ‘‘it never speaks of events inthe system, but only of outcomes of observations uponsystem, implying the existence of external equipment.12

~How, then, do we describe the whole universe, since thcan be no external equipment in this case?!

~vi! Shimony ~Ref. 8!: ‘‘According to the interpretationproposed by Bohr, the change of state is a consequencthe fundamental assumption that the description ofphysical phenomenon requires reference to the experimearrangement.’’

~vii ! Rosenfeld~Ref. 35!: ‘‘the human observer, whom wehave been at pains to keep out of the picture, seems irreibly to intrude into it... .’’

~viii ! Stapp ~Ref. 30!: ‘‘The interpretation of quantumtheory is clouded by the following points:~1! Invalid classi-cal concepts are ascribed fundamental status;~2! The processof measurement is not describable within the frameworkthe theory;~3! The subject-object distinction is blurred;~4!The observed system is required to be isolated in order tdefined, yet interacting to be observed.’’A. Von Neumann’s infinite regress

In this section, we introduce the notion of the Von Nemann regress, or Von Neumann chain, a process that is a


thys-e-e

he

e

s-n-

lt,r-

y

a

i-yn

-ehe

s

e

re

ofytal

st-

f

be

-the

source of the phenomenon of decoherence. Both actucorrespond to the same basic physical process, but the wdecoherence usually refers to its initial stage, when the nber of degrees of freedom involved in the process is srelatively limited. The Von Neumann chain, on the othhand, is more general since it includes this initial stagewell as its continuation, which goes on until it reaches tother extreme where it really becomes paradoxical:Schrodinger cat, the symbol of a macroscopic system, wan enormous number of degrees of freedom, in an impossstate ~Schrodinger uses the word ‘‘ridiculous’’ to describit!. Decoherence in itself is an interesting physical phenoenon that is contained in the Schro¨dinger equation and introduces no particular conceptual problems; the word is retively recent, and so is the observation of the processbeautiful experiments in atomic physics, Ref. 36—for modetails on decoherence, see Sec. V C 2. Since for the momwe are at the stage of a historical introduction of the diculties of quantum mechanics, we will not discuss micscopic decoherence further, but focus the interest on mascopic systems, where serious conceptual difficultiesappear.

Assume that we take a simple system, such as a spinatom, which enters into a Stern–Gerlach spin analyzer. Ifinitial direction of the spin is transverse~with respect to themagnetic field which defines the eigenstates associatedthe apparatus!, the wave function of the atom will split intotwo different wave packets, one which is pulled upwards,other pushed downwards; this is an elementary consequof the linearity of the Schro¨dinger equation. Propagating further, each of the two wave packets may strike a detecwith which they interact by modifying its state as well atheirs; for instance, the incoming spin 1/2 atoms are ionizand produce electrons; as a consequence, the initial cohesuperposition now encompasses new particles. Moreowhen a whole cascade of electrons is produced in photomtipliers, all these additional electrons also become part ofsuperposition. In fact, there is no intrinsic limit in what soobecomes an almost infinite chain: Rapidly, the linearitythe Schro¨dinger equation leads to a state vector which iscoherent superposition of states including a macrosconumber of particles, macroscopic currents and, maybe, poers or recorders which have already printed zeros or onea piece of paper! If we stick to the Schro¨dinger equation,there is nothing to stop this ‘‘infinite Von Neumann regress,’’ which has its seed in the microscopic world but raidly develops into a macroscopic consequence. Can we,instance, accept the idea that, at the end, it is the brain ofexperimenter~who becomes aware of the results! and there-fore a human being, which enters into such a superpositi

Needless to say, no-one has ever observed two contratory results at the same time, and the very notion is not every clear: It would presumably correspond to an experimtal result printed on paper looking more or less like twsuperimposed slides, or a double exposure of a photogrBut in practice we know that we always observe only osingle result in a single experiment; linear superpositiosomehow resolve themselves before they become sufficiemacroscopic to involve measurement apparatuses andselves. It therefore seems obvious13 that a proper theoryshould break the Von Neumann chain, and stop the regwhen ~or maybe before! it reaches the macroscopic worldBut when exactly and how precisely?

660F. Laloe

ore

hola

noh

lyzorvese, asthi

reuBuisdef theutic

ondlem

tee

mct

wiomthein

wh

otaa

ou

nglim

s

poltiebisowO

e

tonc-si-of

n-uchin-

hemlt of

icrmi-nics, itslyr tohes,inghlu-lic

s toveessiontiony?er-

anthatot

e-ininceme

io-on

thelace

c-at.

t tot toopi-

at

eant

androatthecant

B. Wigner’s friend

The question can also be asked differently: In a thewhere the observer plays such an essential role, who istitled to play it? Wigner discusses the role of a friend, whas been asked to perform an experiment, a Stern-Germeasurement for instance~Ref. 37!; the friend may be work-ing in a closed laboratory so that an outside observer willbe aware of the result before he/she opens the door. Whappens just after the particle has emerged from the anaand when its position has been observed inside the labtory, but is not yet known outside? For the outside obserit is natural to consider the whole ensemble of the clolaboratory, containing the experiment as well as his friendthe ‘‘system’’ to be described by a big wave function. Along as the door of the laboratory remains closed andresult of the measurement unknown, this wave function wcontinue to contain a superposition of the two possiblesults; it is only later, when the result becomes known oside, that the wave packet reduction should be applied.clearly, for Wigner’s friend who is inside the laboratory, threasoning is just absurd! He/she will much prefer to consithat the wave function is reduced as soon as the result oexperiment is observed inside the laboratory. We are tback to a point that we already discussed, the absolrelative character of the wave function: Does this contradtion mean that we should consider two state vectors,reduced, one not reduced, during the intermediate periothe experiment? For a discussion by Wigner of the probof the measurement, see Ref. 38.

An unconventional interpretation, sometimes associawith Wigner’s name,14 assumes that the reduction of thwave packet is a real effect which takes place when a humind interacts with the surrounding physical world and aquires some consciousness of its state; in other words,electrical currents in the human brain may be associateda reduction of the state vector of measured objects, by syet unknown physical process. Of course, in this view,reduction takes place under the influence of the experimtalist inside the laboratory and the question of the precedparagraph is settled. But, even if one accepts the someprovocative idea of possible action of the mind~or con-sciousness! on the environment, this point of view does nsuppress all logical difficulties: What is a human mind, whlevel of consciousness is necessary to reduce the wpacket, etc.?

C. Schrodinger’s cat

The famous story of the Schro¨dinger cat~Refs. 39 and 40!illustrates the problem in a different way; it is probably towell known to be described once more in detail here. Letthen just summarize it: The story illustrates how a livicreature could be put into a very strange state, containingand death, by correlation with a decaying radioactive atoand through a Von Neumann chain; the chain includegamma ray detector, electronic amplification, and finallymechanical system that automatically opens a bottle ofsonous gas if the atomic decay takes place. The resuparadox may be seen as an illustration of the following qution: Does an animal such as a cat have the intellectual aties that are necessary to perform a measurement and reall Von Neumann branches into one? Can it perceive its ostate, projecting itself onto one of the alive or dead states?do humans only have access to a sufficient level of introsp


yn-

ch

tatera-r,ds

ell-

t-t,

rhene/-eof

d

an-hethe

en-gat

tve

s

fe,a

ai-

ngs-li-lvenr

c-

tion to become conscious of their own observations, andreduce the wave function? In that case, when the wave fution includes a cat component, the animal could remainmultaneously dead and alive for an arbitrarily long periodtime, a paradoxical situation indeed.

Another view on the paradox is obtained if one just cosiders the cat as a symbol of any macroscopic object; sobjects can obviously never be in a ‘‘blurred’’ state containg possibilities that are obviously contradictory~open andclosed bottle, dead and alive cat, etc.!. Schrodinger considersthis as a ‘‘quite ridiculous case,’’ which emerges from tlinearity of his equation, but should clearly be excluded froany reasonable theory—or at best considered as the resusome incomplete physical description. In Schro¨dinger’swords: ‘‘an indeterminacy originally restricted to the atomdomain becomes transformed into a macroscopic indetenacy.’’ The message is simple: Standard quantum mechais not only incapable of avoiding these ridiculous casesactually provides a recipe for creating them; one obviouneeds some additional ingredients in the theory in orderesolve the Von Neumann regress, select one of its brancand avoid stupid macroscopic superpositions. It is amusto note in passing that Schro¨dinger’s name is associated wittwo contradictory concepts that are actually mutually excsive, a continuous equation of evolution and the symbocat, a limit that the equation should never reach! Needlessay, the limit of validity of the linear equation does not hato be related to the cat itself: The branch selection procmay perfectly take place before the linear superpositreaches the animal. But the real question is that the reducprocess has to take place somewhere, and where exactl

Is this paradox related to decoherence? Not really. Cohence is completely irrelevant for Schro¨dinger, since the cat isactually just a symbol of a macroscopic object that is inimpossible blurred state, encompassing two possibilitiesare incompatible in ordinary life; the state in question is nnecessarily a pure state~only pure states are sensitive to dcoherence! but can also be a statistical mixture. Actually,the story, the cat is never in a coherent superposition, sits blurred state is precisely created by correlation with soparts of the environment~the bottle of poison for instance!;the cat is just another part of the environment of the radactive atom. In other words, the cat is not the seed of a VNeumann chain; it is actually trapped into two~or more! ofits branches, in a tree that has already expanded intomacroscopic world after decoherence has already taken pat a microscopic level~radioactive atom and radiation detetor!, and will continue to expand after it has captured the cDecoherence is irrelevant for Schro¨dinger since his point isnot to discuss the features of the Von Neumann chain, buemphasize the necessity to break it: The question is nohave a coherent or a statistical superposition of macrosccally different states, it is to have no superposition at all!15

So the cat is the symbol of an impossibility, an animal thcan never exist~a Schro¨dinger gargoyle?!, and a tool for a‘‘reductio ad absurdum’’ reasoning that puts into light thlimitations of the description of a physical system bySchrodinger wave function only. Nevertheless, in the receliterature in quantum electronics, it has become moremore frequent to weaken the concept, and to call ‘‘Sch¨-dinger cat~SC!’’ any coherent superposition of states thcan be distinguished macroscopically, independently ofnumbers of degree of freedom of the system. SC statesthen be observed~for instance an ion located in two differen

661F. Laloe

cero

itiaratoth

os—ea

f heoen

fe

hex

athngafatahex

nt

the

Onela

erseaoa

, tvFceiothealI

teovtotetesmi

tiob

umothetat-a

eyat

ky,-theonyeenx-

in-er-ent

o-s,

n,atry,

the

rd-

r toa

edf

isille

rear-

orin-

anyd,iontece

top-the

ionordhe

a

places in a trap!, but often undergo rapid decoherenthrough correlation to the environment. Moreover, the Sch¨-dinger equation can be used to calculate how the instages of the Von Neumann chain take place, and howidly the solution of the equation tends to ramify inbranches containing the environment. Since this use ofwords SC has now become rather common in a subfieldphysics, one has to accept it; it is, after all, just a matterconvention to associate them with microscopic systemany convention is acceptable as long as it does not crconfusion. But it would be an insult to Schro¨dinger to be-lieve that decoherence can be invoked as the solution oinitial cat paradox: Schro¨dinger was indeed aware of thproperties of entanglement in quantum mechanics, a wthat he introduced~and uses explicitly in the article on thcat!, and he was not sufficiently naive to believe that stadard quantum mechanics would predict possible interences between dead and alive cats!

To summarize, the crux of most of our difficulties witquantum mechanics is the following question: What isactly the process that forces Nature to break the regressto make its choice among the various possibilities forresults of experiments? Indeed, the emergence of a siresult in a single experiment, in other words the disappeance of macroscopic superpositions, is a major issue; thethat such superpositions cannot be resolved at any swithin the linear Schro¨dinger equation may be seen as tmajor difficulty of quantum mechanics. As Pearle nicely epresses it, Ref. 12, the problem is to ‘‘explain why eveoccur!’’

D. Unconvincing arguments

We have already emphasized that the invention ofCopenhagen interpretation of quantum mechanics has band remains, one of the big achievements of physics.can admire even more, in retrospect, how early its foundconceived it, at a time when experimental data were retively scarce. Since that time, numerous ingenious expments have been performed, precisely with the hope ofing the limits of this interpretation but, until now, notsingle fact has disproved the theory. It is really a wonderpure logic that has allowed the early emergence of suchintellectual construction.

This being said, one has to admit that, in some casesbrilliant authors of this construction may sometimes hagone too far, pushed by their great desire to convince.instance, authoritative statements have been made coning the absolute necessity of the orthodox interpretatwhich now, in retrospect, seem exaggerated—to sayleast. According to these statements, the orthodox interprtion would give the only ultimate description of physicreality; no finer description would ever become possible.this line of thought, the fundamental probabilistic characof microscopic phenomena should be considered as a prfact, a rule that should be carved into marble and accepforever by scientists. But, now, we know that this is nproven to be true: Yes, one may prefer the orthodox inpretation if one wishes, but this is only a matter of tasother interpretations are still perfectly possible; determiniin itself is not disproved at all. As discussed for instanceRef. 6, and initially clarified by Bell~Refs. 3 and 7! as wellas by Bohm~Refs. 4 and 5!, the ‘‘impossibility proofs’’ putforward by the proponents of the Copenhagen interpretaare logically unsatisfactory for a simple reason: They ar


lp-

eoff

te

is

rd

-r-

-ndeler-ctge

-s

een,e

rs-i-e-

fn

heeorrn-ne

ta-

nrenedtr-;

n

ni-

trarily impose conditions that may be relevant to quantmechanics~linearity!, but not to the theories that they aim tdismiss—any theory with additional variables such asBohm theory, for instance. Because of the exceptional sure of the authors of the impossibility theorems, it tooklong time for the physics community to realize that thwere irrelevant; now, this is more widely recognized so ththe plurality of interpretations is more easily accepted.

III. EINSTEIN, PODOLSKY, AND ROSEN

It is sometimes said that the article by Einstein, Podolsand Rosen~EPR! in Ref. 41 is, by far, that which has collected the largest number of quotations in the literature;statement sounds very likely to be true. There is some irin this situation since, so often, the EPR reasoning has bmisinterpreted, even by prominent physicists! A striking eample is given in the Einstein–Born correspondence~Ref.42! where Born, even in comments that he wrote after Estein’s death, still clearly shows that he never really undstood the nature of the objections raised by EPR. Born won thinking that the point of Einstein was ana priori rejec-tion of indeterminism~‘‘look, Einstein, indeterminism is notso bad’’!, while actually the major concern of EPR was lcality and/or separability~we come back later to these termwhich are related to the notion of space–time!. If giants likeBorn could be misled in this way, no surprise that, later omany others made similar mistakes! This is why, in whfollows, we will take an approach that may look elementabut at least has the advantage of putting the emphasis onlogical structure of the arguments.

A. A theorem

One often speaks of the ‘‘EPR paradox,’’ but the wo‘‘paradox’’ is not really appropriate in this case. For Einstein, the basic motivation was not to invent paradoxes oentertain colleagues inclined to philosophy; it was to buildstrong logical reasoning which, starting from well-definassumptions~roughly speaking: locality and some form orealism!, would lead ineluctably to a clear conclusion~quan-tum mechanics is incomplete, and even: physicsdeterministic!.16 To emphasize this logical structure, we wspeak here of the ‘‘EPR theorem,’’ which formally could bstated as follows:

Theorem: If the predictions of quantum mechanics acorrect (even for systems made of remote correlated pticles) and if physical reality can be described in a local (separable) way, then quantum mechanics is necessarilycomplete: some ‘‘elements of reality’’17 exist in Nature thatare ignored by this theory.

The theorem is valid, and has been scrutinized by mscientists who have found no flaw in its derivation; indeethe logic which leads from the assumptions to the conclusis perfectly sound. It would therefore be an error to repea~aclassical mistake!! ‘‘the theorem was shown by Bohr to bincorrect’’ or, even worse, ‘‘the theorem is incorrect sinexperimental results are in contradiction with it.’’18 Bohrhimself, of course, did not make the error: In his replyEPR ~Ref. 43!, he explains why he thinks that the assumtions on which the theorem is based are not relevant toquantum world, which makes it inapplicable to a discusson quantum mechanics; more precisely, he uses the w‘‘ambiguous’’ to characterize these assumptions, butnever claims that the reasoning is faulty~for more details,see Sec. III B 3!. A theorem which is not applicable in

662F. Laloe

Eautrulo

‘re

in

ef

tsi

ilidhthts

ree

l

s

oeofsin

e.edre

hi

ro

-

siin

waeireh

onwi

-

fic

era-e; ithaton-b-ntsen-met,s, ants

edat

n

essatedthe

-n ofpicred

ndan-uldrse,neisan-the

la-eyme,

ourveser inmeenive

utthe

t cances,gleme-rs.theeof

es-

particular case is not necessarily incorrect: Theorems ofclidean geometry are not wrong, or even useless, becone can also build a consistent non-Euclidean geomeConcerning possible contradictions with experimental reswe will see that, in a sense, they make a theorem even minteresting, mostly because it can then be used within a ‘ductio ad absurdum’’ reasoning.

Good texts on the EPR argument are abundant; forstance, a classic is the wonderful little article by Bell~Ref.33!; another excellent introductory text is, for instance, R44, which contains a complete description of the scheme~inthe particular case where two settings only are used! andprovides an excellent general discussion of many aspecthe problem; for a detailed source of references, see forstance Ref. 45. Most readers are probably already famwith the basic scheme considered, which is summarizeFig. 1: A sourceS emits two correlated particles, whicpropagate toward two remote regions of space whereundergo measurements; the type of these measuremendefined by ‘‘settings,’’ or ‘‘parameters’’19 ~typically orienta-tions of Stern–Gerlach analyzers, often denoteda and b!,which are at the choice of the experimentalists; in eachgion, a result is obtained, which can take only two valusymbolized by61 in the usual notation; finally, we wilassume that, every time both settings are chosen to besame value, the results of both measurements are alwaysame.

Here, rather than trying to paraphrase the good textsEPR with more or less success, we will purposefully takdifferent presentation, based on a comparison, a sortparable. Our purpose is to emphasize a feature of the reaing: The essence of the EPR reasoning is actually nothbut what is usually called ‘‘the scientific method’’ in thsense discussed by Francis Bacon and Claude Bernardthis purpose, we will leave pure physics for botany! Indein both disciplines, one needs rigorous scientific proceduin order to prove the existence of relations and causes, wis precisely what we want to do.

B. Of peas, pods, and genes

When a physicist attempts to infer the properties of micscopic objects from macroscopic observations, ingenuity~inorder to design meaningful experiments! must be combinedwith a good deal of logic~in order to deduce these microscopic properties from the macroscopic results!. Obviously,some abstract reasoning is indispensable, merely becauis impossible to observe with the naked eye, or to takeone’s hand, an electron or even a macromolecule forstance. The scientist of past centuries who, like Mendel,trying to determine the genetic properties of plants, hadactly the same problem: He did not have access to any dobservation of the DNA molecules, so that he had to basereasoning on adequate experiments and on the observatitheir macroscopic outcome. In our parable, the scientistobserve the color of flowers~the ‘‘result’’ of the measure-ment,11 for red,21 for blue! as a function of the conditionin which the peas are grown~these conditions are the ‘‘ex

Fig. 1.


u-sey!tsre-

-

.

ofn-arin

eyare

-s

thethe

naa

on-g

For,s

ch

-

e itn-s

x-ctisof

ll

perimental settings’’a andb, which determine the nature othe measurement!. The basic purpose is to infer the intrinsproperties of the peas~the EPR ‘‘element of reality’’! fromthese observations.

1. Simple experiments; no conclusion yet

It is clear that many external parameters such as tempture, humidity, amount of light, etc., may influence thgrowth of vegetables and, therefore, the color of a flowerseems very difficult in a practical experiment to be sure tall the relevant parameters have been identified and ctrolled with a sufficient accuracy. Consequently, if one oserves that the flowers which grow in a series of experimeare sometimes blue, sometimes red, it is impossible to idtify the reason behind these fluctuations; it may reflect sotrivial irreproducibility of the conditions of the experimenor something more fundamental. In more abstract termcompletely random character of the result of the experimemay originate either from the fluctuations of uncontrollexternal perturbations, or from some intrinsic property ththe measured system~the pea! initially possesses, or evefrom the fact that the growth of a flower~or, more generally,life?! is fundamentally an indeterministic process—needlto say, all three reasons can be combined in any complicway. Transposing the issue to quantum physics leads tofollowing formulation of the following question: Are the results of the experiments random because of the fluctuatiosome uncontrolled influence taking place in the macroscoapparatus, of some microscopic property of the measuparticles, or of some more fundamental process?

The scientist may repeat the ‘‘experiment’’ a thousatimes and even more: If the results are always totally rdom, there is no way to decide which interpretation shobe selected; it is just a matter of personal taste. Of couphilosophical arguments might be built to favor or reject oof them, but from a pure scientific point of view, at thstage, there is no compelling argument for a choice orother. Such was the situation of quantum physics beforeEPR argument.

2. Correlations; causes unveiled

The stroke of genius of EPR was to realize that corretions could allow a big step further in the discussion. Thexploit the fact that, when the settings chosen are the sathe observed results turn out to be always identical; inbotanical analogy, we will assume that our botanist obsercorrelations between colors of flowers. Peas come togethpods, so that it is possible to grow peas taken from the sapod and observe their flowers in remote places. It is thnatural to expect that, when no special care is taken to gequal values to the experimental parameters~temperature,etc.!, nothing special is observed in this new experiment. Bassume that, every time the parameters are chosen tosame values, the colors are systematically the same; whawe then conclude? Since the peas grow in remote plathere is no way that they can be influenced by any sinuncontrolled fluctuating phenomenon, or that they can sohow influence each other in the determination of the coloIf we believe that causes always act locally, we are led tofollowing conclusion: The only possible explanation of thcommon color is the existence of some common propertyboth peas, which determines the color; the property in qu

663F. Laloe

e-ohe

aet

eaenmeserf

anJu

uluelictpeeinyatare

spesthtunoesncwep

art d

way,p

arersoftn

febart

ing,folo

rfuo-w

om-the

sultof

ng

od

dsce

in-entwoa-

forveny

or-e

e-ri-

n aa-ini-logn–

ea-thatin

oftstandon:ctrehisged-stingly

theinex-

inre in

tion may be very difficult to detect directly, since it is prsumably encoded inside some tiny part of a biological mecule, but it is sufficient to determine the results of texperiments.

Since this is the essence of the argument, let us mevery step of the EPR reasoning completely explicit, whtransposed to botany. The key idea is that the nature andnumber of ‘‘elements of reality’’ associated with each pcannot vary under the influence of some remote experimperformed on the other pea. For clarity, let us first assuthat the two experiments are performed at different timOne week, the experimenter grows a pea, then only nweek another pea from the same pod; we assume that pecorrelations of the colors are always observed, withoutspecial influence of the delay between the experiments.after completion of the first experiment~observation of thefirst color!, but still before the second experiment, the resof that future experiment has a perfectly determined vatherefore, there must already exist one element of reaattached to the second pea that corresponds to this faclearly, it cannot be attached to any other object than thefor instance one of the measurement apparatuses, sincobservation of perfect correlations only arises when makmeasurements with peas taken from the same pod. Smetrically, the first pod also had an element of realitytached to it which ensured that its measurement wouldways provide a result that coincides with that of the futumeasurement. The simplest idea that comes to mind iassume that the elements of reality associated with bothare coded in some genetic information, and that the valuethe codes are exactly the same for all peas coming fromsame pod; but other possibilities exist and the precise naand mechanism involved in the elements of reality doreally matter here. The important point is that, since thelements of reality cannot appear by any action at a distathey necessarily also existed before any measurementperformed—presumably even before the two peas were srated.

Finally, let us consider any pair of peas, when theyalready spatially separated, but before the experimentaliscides what type of measurements they will undergo~valuesof the parameters, delay or not, etc.!. We know that, if thedecision turns out to favor time separated measurementsexactly the same parameter, perfect correlations will alwbe observed. Since elements of reality cannot appearchange their values, depending on experiments that areformed in a remote place, the two peas necessarily csome elements of reality with them which completely detmine the color of the flowers; any theory which ignores theelements of reality is incomplete. This completes the pro

It seems difficult not to agree that the method which ledthese conclusions is indeed the scientific method; no tribuor detective would believe that, in any circumstance, percorrelations could be observed in remote places withouting the consequence of some common characteristics shby both objects. Such perfect correlations can then onlyveal the initial common value of some variable attachedthem, which is in turn a consequence of some fluctuatcommon cause in the past~a random choice of pods in a bafor instance!. To express things in technical terms, let usinstance assume that we use the most elaborate technoavailable to build elaborate automata, containing powemodern computers20 if necessary, for the purpose of reprducing the results of the remote experiments: Whatever


l-

kenhe

t,e:xtectyst

lt;

ty—a,thegm--l-

toasoferetee,asa-

ee-

ithsorer-ry-e.oalcte-rede-og

rgyl

e

do, we must ensure that, somehow, the memory of each cputer contains the encoded information concerning allresults that it might have to provide in the future~for anytype of measurement that might be made!.

To summarize this section, we have shown that each reof a measurement may be a function of two kindsvariables:21

~i! intrinsic properties of the peas, which they carry alowith them,

~ii ! the local setting of the experiment~temperature, hu-midity, etc.!; clearly, a given pair that turned out tprovide two blue flowers could have provided reflowers in other experimental conditions.

We may also add the following.~iii ! The results are well-defined functions; in other wor

no fundamentally indeterministic process takes plain the experiments.

~iv! When taken from its pod, a pea cannot ‘‘knowadvance’’ to which sort of experiment it will be submitted, since the decision may not yet have bemade by the experimenters; when separated, thepeas therefore have to take with them all the informtion necessary to determine the color of flowersany kind of experimental conditions. What we hashown actually is that each pea carries with it as maelements of reality as necessary to provide ‘‘the crect answer’’22 to all possible questions it might bsubmitted to.

3. Transposition to physics

The starting point of EPR is to assume that quantum mchanics provides correct predictions for all results of expements; this is why we have built the parable of the peas iway that exactly mimics the quantum predictions for mesurements performed on two spin 1/2 particles for sometial quantum state: The red/blue color is obviously the anato the result that can be obtained for a spin in a SterGerlach apparatus, and the parameters~or settings! are analo-gous to the orientation of these apparatuses~rotation aroundthe axis of propagation of the particles!. Quantum mechanicspredicts that the distance and times at which the spin msurements are performed are completely irrelevant, sothe correlations will remain the same if they take placevery remote places.

Another ingredient of the EPR reasoning is the notion‘‘elements of reality;’’ EPR first remark that these elemencannot be found bya priori philosophical considerations, bumust be found by an appeal to results of experimentsmeasurements. They then propose the following criteri‘‘if, without in any way disturbing a system, we can prediwith certainty the value of a physical quantity, then theexists an element of physical reality corresponding to tphysical quantity.’’ In other words, certainty cannot emerfrom nothing: An experimental result that is known in avance is necessarily the consequence of some pre-exiphysical property. In our botanical analogy, we implicitmade use of this idea in the reasoning of Sec. III B 2.

A last, but essential, ingredient of the EPR reasoning isnotion of space–time and locality: The elements of realityquestion are attached to the region of space where theperiment takes place, and they cannot vary suddenly~or evenless appear! under the influence of events taking placevery distant regions of space. The peas of the parable we

664F. Laloe

cyminsecaingonnlir

yabntio

th

c

cinithn

nmc

eyerro

tehee

sm

uncle

ajthaoe

tiaor

rlyex-red

p--the

re-theat-

theex-as atoingati-.ra-s an aandbelo-

hnsinent

tsmscs--’’eallhis

b-ughar-is-re-n

r

phvecul-

fact not so much the symbol of some microscopic objeelectrons, or spin 1/2 atoms for instance. Rather, they sbolize regions of space where we just know that ‘‘somethis propagating;’’ it can be a particle, a field, or anything elwith absolutely no assumption on its structure or physidescription. Actually, in the EPR quotation of the precedparagraph, one may replace the word ‘‘system’’ by ‘‘regiof space,’’ without altering the rest of the reasoning. Omay summarize the situation by saying that the basic beof EPR is that regions of space can contain elements ofality attached to them~attaching distinct elements of realitto separate regions of space is sometimes called ‘‘separity’’ ! and that they evolve locally. From these assumptioEPR prove that the results of the measurements are funcof

~i! intrinsic properties of the spins that they carry withem ~the EPR elements of reality! and

~ii ! of course, also of the orientations of the Stern–Gerlaanalyzers.

In addition, they show the following.~iii ! The functions giving the results are well-defined fun

tions, which implies that no indeterministic process is takplace; in other words, a particle with spin carries along wit all the information necessary to provide the result to apossible measurement.

~iv! Since it is possible to envisage future measuremeof observables that are called ‘‘incompatible’’ in quantumechanics, as a matter of fact, incompatible observablessimultaneously have a perfectly well-defined value.

Item ~i! may be called the EPR-1 result: Quantum mchanics is incomplete~EPR require from a complete theorthat ‘‘every element of physical reality must have a countpart in the physical theory’’!; in other words, the state vectomay be a sufficient description for a statistical ensemblepairs, but for one single pair of spins, it should be compleby some additional information; in other words, inside tensemble of all pairs, one can distinguish between subsembles with different physical properties. Item~iii ! may becalled EPR-2, and establishes the validity of determinifrom a locality assumption. Item~iv!, the EPR-3 result,shows that the notion of incompatible observables is not fdamental, but just a consequence of the incomplete charaof the theory; it actually provides a reason to reject compmentarity. Curiously, EPR-3 is often presented as the mEPR result, sometimes even with no mention of the two oers; actually, the rejection of complementarity is almost mginal or, at least, less important for EPR than the proofincompleteness. In fact, in all that follows in this article, wwill only need EPR-1,2.

Niels Bohr, in his reply to the EPR article~Ref. 43!, statedthat their criterion for physical reality contains an essenambiguity when it is applied to quantum phenomena. A mextensive quotation of Bohr’s reply is the following:

‘‘The wording of the above mentioned crite-rion ~the EPR criterion for elements of reality!...contains an ambiguity as regards the expression‘without in any way disturbing a system.’ Ofcourse there is in a case like that considered~byEPR! no question of a mechanical disturbance ofthe system under investigation during the lastcritical stage of the measuring procedure. Buteven at this stage there is essentially the questionof an influence of the very conditions which de-


t,-

g,l

eefe-

il-s,ns

h

-g

y

ts

an

-

-

fd

n-

-ter-

or-

r-f

le

fine the possible types of predictions regardingthe future behavior of the system... the quantumdescription may be characterized as a rationalutilization of all possibilities of unambiguous in-terpretation of measurements, compatible withthe finite and uncontrollable interactions betweenthe objects and the measuring instruments in thefield of quantum theory.’’

Indeed, in Bohr’s view, physical reality cannot be propedefined without reference to a complete and well-definedperiment. This includes not only the systems to be measu~the microscopic particles!, but also all the measurement aparatuses: ‘‘these~experimental! conditions must be considered as an inherent element of any phenomenon to whichterm physical reality can be unambiguously applied.’’ Thefore EPR’s attempt to assign elements of reality to one ofspins only, or to a region of space containing it, is incompible with orthodox quantum mechanics23—even if the regionin question is very large and isolated from the rest ofworld. Expressed differently, a physical system that istended over a large region of space is to be consideredsingle entity, within which no attempt should be madedistinguish physical subsystems or any substructure; tryto attach physical reality to regions of space is then automcally bound to failure. In terms of our Leitmotiv of SecI A 3, the difference between ordinary space and configution space, we could say the following: The system hasingle wave function for both particles that propagates iconfiguration space with more than three dimensions,this should be taken very seriously; no attempt shouldmade to come back to three dimensions and implementcality arguments in a smaller space.

Bohr’s point of view is, of course, not contradictory witrelativity, but since it minimizes the impact of basic notiosuch as space–time, or events~a measurement processquantum mechanics is not local; therefore it is not an evstricto sensu!, it does not fit very well with it. One could addthat Bohr’s article is difficult to understand; many physicisadmit that a precise characterization of his attitude, in terfor instance of exactly what traditional principles of physishould be given up, is delicate~see, for example, the discussion of Ref. 8!. In Pearle’s words: ‘‘Bohr’s rebuttal was essentially that Einstein’s opinion disagreed with his own~Ref. 46!. It is true that, when scrutinizing Bohr’s texts, onis never completely sure to what extent he fully realizedthe consequences of his position. Actually, in most ofreply to EPR inPhysical Review~Ref. 43!, he just repeats theorthodox point of view in the case of a single particle sumitted to incompatible measurements, and even goes throconsiderations that are not obviously related to the EPRgument, as if he did not appreciate how interesting the dcussion becomes for two remote correlated particles; thelation to locality is not explicitly discussed, as if this was aunimportant issue~while it was the starting point of furtheimportant work, the Bell theorem for instance24!. The precisereply to EPR is actually contained in only a short paragraof this article, from which the quotations given above habeen taken. Even Bell confessed that he had strong diffities understanding Bohr~‘‘I have very little idea what thismeans... .’’ See the appendix of Ref. 33!.

665F. Laloe

en-

elan

th

uriithacono

bsre

ueme

thf

-siay

iblf

agteof

iic

arsenwioavPRe

em

peb

dn

Ifts

mWe

ualre-

, butany

thee

t ised

nyion

hgpins.

canlt

spler

s

fays

IV. QUANTITATIVE THEOREMS: BELL,GREENBERGER–HORNE–ZEILINGER, HARDY,BELL –KOCHEN –SPECKER

The Bell theorem, Ref. 47, may be seen in many differways. In fact, Bell initially invented it as a logical continuation of the EPR theorem: The idea is to take completseriously the existence of the EPR elements of reality,introduce them into the mathematics with the notationl; onethen proceeds to study all possible kinds of correlationscan be obtained from the fluctuations of thel’s, making thecondition of locality explicit in the mathematics~locality wasalready useful in the EPR theorem, but not used in eqtions!. As a continuation of EPR, the reasoning necessadevelops from a deterministic framework and deals wclassical probabilities; it studies in a completely general wall kinds of correlation that can be predicted from the flutuations in the past of some classical common cause—ifprefers, from some uncertainty concerning the initial statethe system. This leads to the famous inequalities. But suquent studies have shown that the scope of the Bell theois not limited to determinism; for instance, thel’s may in-fluence the results of future experiments by fixing the valof probabilities of the results, instead of these results theselves~see Appendix A!. We postpone the discussion of thvarious possible generalizations to Sec. IV A 4 and, formoment, we just emphasize that the essential conditionthe validity of the Bell theorem is locality: All kinds of fluctuations can be assumed, but their effect must affect phyonly locally. If we assume that throwing dice in Paris minfluence physical events taking place in Tokyo, or evenother galaxies, the proof of the theorem is no longer possiFor nonspecialized discussions of the Bell theorem, seeinstance Refs. 33, 44, 48, and 49.

A. Bell inequalities

The Bell inequalities are relations satisfied by the avervalues of product of random variables that are correlaclassically~their correlations arise from the fluctuationssome common cause in the past, as above for the peas!. Aswe will see, the inequalities are especially interestingcases where they are contradictory with quantum mechanone of these situations occurs in the EPRB~B for Bohm, Ref.50! version of the EPR argument, where two spin 1/2 pticles undergo measurements. This is why we begin thistion by briefly recalling the predictions of quantum mechaics for such a physical system—but the only ingredientneed from quantum mechanics at this stage is the predictconcerning the probabilities of results. Then we again lestandard quantum mechanics and come back to the EBell argument, discuss its contradictions with quantum mchanics, and finally emphasize the generality of the theor

1. Two spins in a quantum singlet state

We assume that two spin 1/2 particles propagate in opsite directions after leaving a source which has emitted thin a singlet spin state. Their spin state is then described

uC&51

&@ u1,2.2u2,1.#. ~1!

When they reach distant locations, they are then submittespin measurements, with Stern–Gerlach apparatuses oriealong anglesa andb around the direction of propagation.u is the angle betweena andb, quantum mechanics predic


t

yd

at

a-ly

y-efe-m

s-

eor

cs

ne.or

ed

ns;

-c--ense–

-.

o-my

toted

that the probability for a double detection of results11, 11~or of 21, 21! is

P1,15P2,25sin2 u, ~2!

while the probability of two opposite results is

P1,25P2,15cos2 u. ~3!

This is all that we want to know, for the moment, of quantumechanics: probability of the results of measurements.note in passing that, ifu50 ~when the orientations of themeasurement apparatuses are parallel!, the formulas predictthat one of the probabilities vanishes, while the other is eqto one; therefore the condition of perfect correlationsquired by the EPR reasoning is fulfilled~in fact, the resultsof the experiments are always opposed, instead of equalit is easy to convince oneself that this does not haveimpact on the reasoning!.

2. Proof

We now come back to the line of the EPR theorem. Inframework of strict deterministic theories, the proof of thBell theorem is the matter of a few lines; the longest paractually the definition of the notation. Following Bell, wassume thatl represents all ‘‘elements of reality’’ associatewith the spins; it should be understood thatl is only a con-cise notation which may summarize a vector with macomponents, so that we are not introducing any limitathere. In fact, one can even include inl components whichplay no special role in the problem; the only thing whicmatters is thatl does contain all the information concerninthe results of possible measurements performed on the sWe use another classical notation,A andB, for these results,and small lettersa andb for the settings~parameters! of thecorresponding apparatuses. ClearlyA andB may depend, notonly on l, but also on the settingsa and b; neverthelesslocality requests thatb has no influence on the resultA ~sincethe distance between the locations of the measurementsbe arbitrarily large!; conversely,a has no influence on resuB. We therefore callA(a,l) andB(b,l) the correspondingfunctions~their values are either11 or 21!.

In what follows, it is sufficient to consider two directiononly for each separate measurement; we then use the simnotation:

A~a,l!5A, A~a8,l!5A8 ~4!

and

B~b,l!5B, B~b8,l!5B8. ~5!

For each pair of particles,l is fixed, and the four numberhave well-defined values~which can only be61!. WithEberhard, Ref. 51, we notice that the product

M5AB1AB82A8B1A8B85~A2A8!B1~A1A8!B8~6!

is always equal to either12, or to22; this is because one othe brackets on the right-hand side of this equation alwvanishes, while the other is62. Now, if we take the averagevalue of M over a large number of emitted pairs~averageover l!, since each instance ofM is limited to these twovalues, we necessarily have

22< ^M & <12. ~7!

666F. Laloe

echau

onatigt:de

inanPn

eeaofw

eer

Nit

ient

, f

gfead

nab

fs-ith

mon

o

rea

iesbe

butre-

hey, as

llyy

cingn-f

ndm-ateonense,

, orentadmo-ee

heatti-

s ofimein-that

eralrce

ringr-lly,

ostn-e

e

re-n

ionayof

pa-tec-3

onee

ting

el-

o-

This is the so-called BCHSH form~Ref. 52! of the Belltheorem: The average values of all possible kinds of msurements that provide random results, whatever the menism behind them may be~as long as the randomness is locand arises from the effect of some common fluctuating cain the past!, necessarily obey this strict inequality.

3. Contradiction with quantum mechanics and withexperiments

The generality of the proof is such that one could reasably expect that any sensible physical theory will automcally give predictions that also obey this inequality; the bsurprise was to realize that quantum mechanics does noturns out that, for some appropriate choices of the fourrectionsa,a8,b,b8 ~the precise values do not matter for thdiscussion here!, the inequality is violated by a factor&,which is more than 40%. Therefore, the EPR–Bell reasonleads to a quantitative contradiction with quantum mechics; indeed, the latter is not a local realistic theory in the Esense. How is this contradiction possible, and how careasoning that is so simple be incorrect within quantum mchanics? The answer is the following: What is wrong, if wbelieve quantum mechanics, is to attribute well-defined vues A,A8,B,B8 to each emitted pair; because only twothem at maximum can be measured in any experiment,cannot speak of these four quantities, or reason on theven as unknown quantities. As nicely emphasized by Pin an excellent short article~Ref. 53!, ‘‘unperformed experi-ments have no result,’’ that is all!

Wheeler expresses a similar idea when he writes: ‘‘elementary quantum phenomenon is a phenomenon untila recorded phenomenon’’~Ref. 54!. As for Wigner, he em-phasizes in Ref. 55 that the proof of the Bell inequalitrelies on a very simple notion: the number of categories iwhich one can classify all pairs of particles.25 Each categoryis associated with well-defined results of measurementsthe various choices of the settingsa and b that are consid-ered; in any sequence of repeated experiments, each catewill contribute with some given weight, its probability ooccurrence, which it has to positive or zero. Wigner thnotes that, if one introduces the notion of locality, each cegory becomes the intersection of a subensemble thatpends ona only, by another subensemble that depends obonly. This operation immediately reduces the number of cegories: In a specific example, he shows that their numreduces from 49 to (23)2526. The mathematical origin othe Bell inequalities lies precisely in the possibility of ditributing all pairs into this smaller number of categories, wpositive probabilities.

A general way to express the Bell theorem in logical teris to state that the following system of three assumpti~which could be called the EPR assumptions! is self-contradictory:

~1! validity of their notion of ‘‘elements of reality,’’~2! locality,~3! the predictions of quantum mechanics are always c

rect.

The Bell theorem then becomes a useful tool to build a ‘‘ductio ad absurdum’’ reasoning: It shows that, amongthree assumptions, one~at least! has to be given up. Themotivation of the experimental tests of the Bell inequalitwas precisely to check if it was not the third which should


a-a-

lse

-i-

Iti-

g-

Ra-

l-

em,es

ois

so

or

ory

nt-e-

t-er

ss

r-

-ll

abandoned. Maybe, after all, the Bell theorem is nothingan accurate pointer toward exotic situations where the pdictions of quantum mechanics are so paradoxical that tare actually wrong? Such was the hope of some theoristswell as the exciting challenge to experimentalists.

Experiments were performed in the seventies, initiawith photons~Refs. 56 and 57! where they already gave verclear results, as well as with protons~Ref. 58!; in the eight-ies, they were made more and more precise and convin~Ref. 59—see also Ref. 60!; ever since, they have been costantly improved~see for instance Ref. 61, but the list oreferences is too long to be given here!; all these results haveclearly shown that, in this conflict between local realism aquantum mechanics, the latter wins completely. A fair sumary of the situation is that, even in these most intricsituations invented and tested by the experimentalists, nohas been able to disprove quantum mechanics. In this sewe can say that Nature obeys laws which are nonlocalnonrealist, or both. It goes without saying that no experimin physics is perfect, and it is always possible to inventhoc scenarios where some physical processes, for thement totally unknown, ‘‘conspire’’ in order to give us thillusion of correct predictions of quantum mechanics—wcome back to this point in Sec. V A—but the quality and tnumber of the experimental results does not make thistude very attractive intellectually.

4. Generality of the theorem

We have already mentioned that several generalizationthe Bell theorem are possible; they are at the same tmathematically simple and conceptually interesting. Forstance, in some of these generalizations, it is assumedthe result of an experiment becomes a function of sevfluctuating causes: the fluctuations taking place in the souas usual, but also fluctuations taking place in the measuapparatuses~Ref. 62!, and/or perturbations acting on the paticles during their motion toward the apparatuses; actuaeven fundamentally indeterministic~but local! processesmay influence the results. The two former cases are almtrivial since they just require the addition of more dimesions to the vector variablel; the latter requires replacing thdeterministic functionsA and B by probabilities, but this isalso relatively straightforward, Ref. 49~see also the footnotein Ref. 62 and Appendix A!. Moreover, one should realizthat the role of theA and B functions is just to relate theconditions of production of a pair of particles~or of theirpropagation! to their behavior when they reach the measument apparatuses~and to the effects that they produce othem!; they are, so to say, solutions of the equation of motwhatever these are. The important point is that they mperfectly include, in a condensed notation, a large varietyphysical phenomena: propagation of point particles, progation of one or several fields from the source to the detors ~see for instance the discussion in Sec. 4 of Ref. 3!,particles, and fields in interaction, or whatever processmay have in mind ~even random propagations can bincluded!—as long as they do not depend on the other set~A is supposed to be a function ofa, not of b!. The exactmathematical form of the equations of propagation is irrevant; the essential thing is that the functions exist.

Indeed, what really matters for the proof of the Bell therem is the dependence with respect to the settingsa and b:The functionA must depend ona only, whileB must dependon b only. Locality expressed mathematically in terms ofa

667F. Laloe

eothb

ro

llyssetesa

th

ganetht

-n

osthe-a

tivn

5antiv

reontio.e

fpal-

thinndinb’’

damew

et-innend

th

ceticale

ics:at-

n-la-a-

t of. In

bi-wean-

m-

tro-

itionofthisatts,

he

the

t

is

i-

arederre-

andb is the crucial ingredient. For instance we could, if wwished, assume that the resultA of one measurement is alsa function of fluctuating random variables attached toother apparatus, which introduces a nonlocal process;this does not create any mathematical problem for the p~as long as these variables are not affected by settingb!. Onthe other hand, ifA becomes a function ofa and b ~and/orthe same forB!, it is easy to see that the situation is radicachanged: In the reasoning of Sec. IV A 2 we must now asciate eight numbers to each pair~since there are two resultto specify for each of the four different combinations of stings!, instead of four, so that the proof miserably collapsAppendix A gives another concrete illustration showing thit is locality, not determinism, which is at stake; see alsoappendix of Ref. 49.

Needless to say, the independence ofA of b does not meanthat the result observed on one side,A, is independent of theoutcome at the other side,B: One should not confuse settinand outcome dependencies! It is actually clear that, intheory, the correlations would disappear if outcome depdence was totally excluded. We should also mention thatsetting dependence is subject to some constraints, iftheory is to remain compatible with relativity. If, for instance, the probability of observation of the results on oside, which is a sum of probabilities over the various psible outcomes on the other side, was still a function ofother setting, one would run into incompatibility; this is bcause one could use the device to send signals withoutfundamental delay, thus violating the constraints of relaity. See Refs. 63 and 64 for a discussion in terms of ‘‘strolocality’’ and ‘‘predictive completeness’’~or ‘‘parameter in-dependence’’ and of ‘‘outcome independence’’ in Ref. 6!.Appendix D discusses how the general formalism of qutum mechanics manages to ensure compatibility with relaity.

An interesting generalization of the Bell theorem, whetime replaces the settings, has been proposed by FransRef. 66 and implemented in experiments for an observaof a violation of the Bell inequalities~see for instance Ref67!; another generalization shows that a violation of the Binequalities is not limited to a few quantum states~singlet forinstance!, but includes all states that are not products, Re68 and 69. For a general discussion of the conceptual imof a violation of the inequalities, we refer to the book colecting Bell’s articles~Ref. 7!.

We wish to conclude this section by emphasizing thatBell theorem is much more general than many people thAll potential authors on the subject should think twice aremember this carefully before taking their pen and senda manuscript to a physics journal: Every year a large numof them is submitted, with the purpose of introducing ‘‘newways to escape the constraints of the Bell theorem, an‘‘explain’’ why the experiments have provided results thare in contradiction with the inequalities. According to thethe nonlocal correlations would originate from some nsort of statistics, or from perturbations created by cosmrays, gas collisions with fluctuating impact parameters,The imagination is the only limit of the variety of the processes that can be invoked, but we know from the beginnthat all these attempts are doomed to failure. The situatioanalogous to the attempts of past centuries to invent ‘‘ppetuum mobile’’ devices: Even if some of these inventiowere extremely clever, and if it is sometimes difficult to finthe exact reason why they cannot work, it remains true


eutof

o-

-.te

yn-e

he

e-e

ny-g

--

inn

ll

s.ct

ek.

ger

tot,

icc.

gisr-s

at

the law of energy conservation allows us to know at onthat they cannot. In the same way, some of these statis‘‘Bell beating schemes’’ may be extremely clever, but wknow that the theorem is a very general theorem in statistIn all situations that can be accommodated by the mathemics of the l’s and theA and B functions ~and there aremany!!, it is impossible to escape the inequalities. No, nolocal correlations cannot be explained cheaply; yes, a viotion of the inequalities is therefore a very, very, rare sitution. In fact, until now, it has never been observed, excepcourse in experiments designed precisely for this purposeother words, if we wanted to build automata including artrarily complex mechanical systems and computers,could never mimic the results predicted by quantum mechics ~at least for remote measurements!; this will remain im-possible forever, or at least until completely different coputers working on purely quantum principles are built.26

B. Hardy’s impossibilities

Another scheme of the same conceptual type was induced recently by Hardy~Ref. 70!; it also considers twoparticles but it is nevertheless completely different sinceinvolves, instead of mathematical constraints on correlatrates, the very possibility of occurrence for some typeevents—see also Ref. 71 for a general discussion ofinteresting contradiction. As in Sec. IV A 2, we assume ththe first particle may undergo two kinds of measuremencharacterized by two valuesa and a8 of the first setting; ifwe reason as in the second half of Sec. IV A 2, within tframe of local realism, we can callA andA8 the correspond-ing results. Similar measurements can be performed onsecond particle, and we callB andB8 the results.

Let us now consider three types of situations:

~i! settings without prime—we assume that the resulA51, B51 is sometimes obtained,

~ii ! one prime only—we assume that the ‘‘double one’’impossible, in other words that one never getsA51,B851, and neverA851, B51 either,

~iii ! double prime settings—we assume that ‘‘double mnus one’’ is impossible, in other words thatA8521, B8521 is never observed.

A closer inspection shows that these three assumptionsin fact incompatible. To see why, let us for instance consithe logical scheme of Fig. 2, where the upper part corsponds to the possibility opened by statement~i!; statement~ii ! then implies that, ifA51, one necessarily hasB8521,

Fig. 2.

668F. Laloe

nnt-ntbl

sowth

wny

th

e

th

b

th

-at

he

le

heeno

toichs

nto

em:our-nt.b-ureo-umngsersr a73.

allyingtheIten-

which explains the first diagonal in the figure; the secodiagonal follows by symmetry. Then we see that all evecorresponding to the resultsA5B51 also necessarily correspond toA85B8521, so that a contradiction with stateme~iii ! appears: The three propositions are in fact incompatiA way to express it is to say that the ‘‘sometimes’’ of~i! iscontradictory with the ‘‘never’’ of proposition~iii !.

But it turns out that quantum mechanics does allow amultaneous realization of all three propositions! To see hlet us for instance consider a two-spin state vector ofform:

uF&5au1,2&1bu2,1&1gu1,1& ~8!

where theu6,6& refer to eigenstates ofA8 andB8 ~Note: axisOz is chosen as the direction of measurement associatedprimed operators!. From the beginning, the absence of auC& component onu2,2& ensures that proposition~iii ! is true.As for the measurements without prime, we assume thatare both performed along a direction in the planexOz thatmakes an angle 2u with Oz; the eigenstate with eigenvalu11 associated in the single-spin state is then merely

cosuu1&1sinuu2&. ~9!

The first state excluded by proposition~ii ! ~diagonal in Fig.2! is then the two-spin state:

cosuu1,1&1sinuu2,1& ~10!

while the second is:

cosuu1,1&1sinuu2,1& ~11!

so that the two exclusion conditions are equivalent tofollowing conditions:

a sinu1g cosu5b sinu1g cosu50 ~12!

or, within a proportionality coefficient:

a5b52g cotu. ~13!

This arbitrary coefficient may be used to writeuC& in theform:

uC&52cosu~ u1,2&1u2,1&)1sinuu1,1&. ~14!

The last thing to do is to get the scalar product of this ketthat where the two spins are in the state~9!; we get thefollowing result:

2sinu cos2 u. ~15!

The final step is to divide this result by the square ofnorm of ket ~14! in order to obtain the probability of theprocess considered in~iii !; this is a straightforward calculation ~see Appendix B!, but here we just need to point out ththe probability is not zero; the precise value of itsu maxi-mum found in Appendix B is about 9%. This proves that tpair of results considered in proposition~i! can sometimes beobtained together with~ii ! and ~iii !: Indeed, in 9% of thecases, the predictions of quantum mechanics are in compcontradiction with those of a local realist reasoning.

An interesting aspect of the above propositions is that tcan be generalized to an arbitrary number of measurem~Ref. 72!; it turns out that this allows a significant increasethe percentage of ‘‘impossible events’’~impossible withinlocal realism! predicted by quantum mechanics—from 9%almost 50%! The generalization involves a chain, whkeeps the two first lines~i! and ~ii ! unchanged, and iteratethe second in a recurrent way, by assuming that:


ds

e.

i-,e

ith

ey

e

y

e

te

yts

f

~iii ! for measurements of the type (a8,b9) or (a9,b8), onenever gets opposite results,27

~iv! for measurements of the type (a9,b-) or (a-,b9),one never gets opposite results, etc.,

~n! finally, for measurement of the type (an,bn), onenever gets21 and21.

The incompatibility proof is very similar to that giveabove; it is summarized in Fig. 3. In both cases, the wayresolve the contradiction is the same as for the Bell theorIn quantum mechanics, it is not correct to reason on all fquantitiesA, A8, B, andB8, even as quantities that are unknown and that could be determined in a future experimeThis is simply because, with a given pair of spins, it is oviously impossible to design an experiment that will measall of them: They are incompatible. If we insisted on intrducing similar quantities to reproduce the results of quantmechanics, since four experimental combinations of settiare considered, we would have to consider eight numbinstead of four, as already discussed in Sec. IV A 4. Fodiscussion of nonlocal effects with other states, see Ref.

C. GHZ equality

For many years, everyone thought that Bell had basicexhausted the subject by considering all really interestsituations, in other words that two-spin systems providedmost spectacular quantum violations of local realism.therefore came as a surprise to many when in 1989 Gre

Fig. 3.

669F. Laloe

ttho

aesin

eears/2t

th

mbnc

th

e

d

liseshiin,ilshat

he

inimndw

duct

of

ra-do

re

her-theichrct ofentper-picved,

rthe-lec-

ady-e.enhere

intokes

thetheyno

hinump-t-oteorle-taln-ute,

allinghreech oflity.oreuct

ure-foroseorsr. Ittup

lity

pre-

berger, Horne, and Zeilinger~GHZ! showed that this is notrue: As soon as one considers systems containing moretwo correlated particles, even more dramatic violationslocal realism become possible in quantum mechanics,even without involving inequalities. Here, we limit ourselvto the discussion of three particle systems, as in the origarticles~Refs. 74 and 75!, but generalization toN particles ispossible; see for instance Sec. V C 1 or Ref. 76. While R75 discussed the properties of three correlated photons,emitted through two pinholes and impinging beam splittewe will follow Ref. 77 and consider a system of three 1spins~external variables play no role here!; we assume thathe system is described by the quantum state:

uC&51

&@ u1,1,1&2u2,2,2&], ~16!

where theu6& states are the eigenstates of the spins alongOzaxis of an orthonormal frameOxyz. We now calculate thequantum probabilities of measurements of the spinss1,2,3 ofthe three particles, either along directionOx, or along direc-tion Oy, which is perpendicular. More precisely, we assuthat what is measured is not individual spin components,only the product of three of these components, for instas1y3s2y3s3x . A straightforward calculation~see Appen-dix C! shows that

P~s1y3s2y3s3x51!511,

P~s1x3s2y3s3y51!511, ~17!

P~s1y3s2x3s3y51!511.

In fact, the state vector written in~16! turns out to be acommon eigenstate to all three operator products, soeach of them takes a value11 that is known before themeasurement.28 Now, if we consider the product of threspin components alongOx, it is easy to check~Appendix C!that the same state vector is also an eigenstate of the prooperators1x3s2x3s3x , but now with eigenvalue21, sothat

P~s1x3s2x3s3x521!51. ~18!

This time the result is21, with probability 1, that is withcertainty.

Let us now investigate the predictions of a local reaEPR type point of view in this kind of situation. Since thquantum calculation is so straightforward, it may seem uless: Indeed, no one before GHZ suspected that anytinteresting could occur in such a simple case, where thetial state is an eigenstate of all observables consideredthat the results are perfectly certain. But, actually, we wsee that a complete contradiction emerges from this analyThe local realist reasoning is a simple generalization of tgiven in Sec. IV A 2; we callAx,y the results that the firsspin will give for a measurement, either alongOx or Oy;similar lettersB andC are used for the measurement on ttwo other spins. From the three equalities written in~17! wethen get:

AyByCx51, AxByCy51, AyBxCy51. ~19!

Now, if we assume that the observations of the three spare performed in three remote regions of space, localityplies that the values measured for each spin should be ipendent of the type of observation performed on the tother spins. This means that the same values ofA, B, andC


anf

nd

al

f.ch,

e

eute

at

uct

t

e-ngi-solis!t

s-e-o

can be used again for the experiment where the threeOxcomponents are measured: The result is merely the proAxBxCx . But, since the squaresAy

2, etc., are always equal to11, we can obtain this result by multiplying all three partsEq. ~19!, which provides

AxBxCx511. ~20!

But equality~18! predicts the opposite sign!Here we obtain a contradiction that looks even more d

matic than for the Bell inequalities: The two predictionsnot differ by some significant fraction~about 40%!, they arejust completely opposite. In addition, all fluctuations aeliminated since all of the results~the products of the threecomponents! are perfectly known before measurement: T100% contradiction is obtained with 100% certainty! Unfotunately, this does not mean that, experimentally, tests ofGHZ equality are easy. Three particles are involved, whmust be put in state~16!, surely a nontrivial task; moreoveone has to design apparatuses that measure the produthree spin components. To our knowledge, no experimanalogous to the Bell inequality experiments has beenformed on the GHZ equality yet, at least with macroscodistances; only microscopic analogues have been obserin nuclear magnetic resonance experiments~Ref. 78!—forrecent proposals, see for instance Refs. 79 and 80. Neveless, constant progress in the techniques of quantum etronics is taking place, and GHZ entanglement has alrebeen observed~Refs. 81 and 82!, so that one gets the impression that a full experiment is not too far away in the futur

In a GHZ situation, how precisely is the conflict betwethe reasoning above and quantum mechanics resolved? Tare different stages at which this reasoning can be putquestion. First, we have assumed locality, which here tathe form of noncontextuality~see Sec. IV D!: Each of theresults is supposed to be independent of the nature ofmeasurements that are performed on the others, becausetake place in remote regions of space. Clearly, there isspecial reason why this should necessarily be true witquantum mechanics. Second, we have also made asstions concerning the nature of the ‘‘elements of reality’’ atached to the particles. In this respect, it is interesting to nthat the situation is very different from the EPR–BellHardy cases: Bohr could not have replied that different ements of reality should be attached to different experimensetups! In the GHZ argument, it turns out that all four quatum operators corresponding to the measurements commso that there is in principle no impossibility of measuringof them with a single setup. But the local realist reasonalso assumes that a measurement of the product of toperators is equivalent to a separate measurement of eathem, which attributes to them separate elements of reaIn the formalism of quantum mechanics, the question is msubtle. It turns out that the measurement of a single prodof commuting operators is indeed equivalent to the measment of each of them; but this is no longer the caseseveral product operators, as precisely illustrated by thintroduced above: Clearly, all six spin component operatappearing in the formulas do not commute with each otheis therefore impossible to design a single experimental seto have access to all six quantitiesAx,y , Bx,y , andCx,y thatwe have used in the local realist proof.29

When the measurements are imperfect, the GHZ equacan give rise to inequalities~as in the BCHSH theorem!, asdiscussed in Refs. 75 and 83; the latter reference also

670F. Laloe

the

a

sto

ssth-tsde

f aelle

opopmT

foe

an

e-

1tha

eth

mnyre

embto

nS

–tioeyw

ak-op-sur-

pinre-

ach

r 0.les,

ili-

forisgo-

ible

ne

ed,

re

htllos-el-

ca-byree.blein

lme

tion

psises:of

an-ns

twoor

sents a generalization to an arbitrary numberN of particles;in the same line, Ref. 76 provides a discussion ofN-particle correlation function with varying angles for thanalyzers, which we partially reproduce in Sec. V C 1.

D. Bell–Kochen–Specker; contextuality

Another theorem was introduced also by Bell, Ref. 3,well as~independently and very shortly after! by Kochen andSpecker~Ref. 84!, hence the name ‘‘BKS theorem’’ that ioften used for it. This theorem is not particularly relatedlocality, as opposed to those that we have already discuin the preceding subsections. It is actually related to anonotion, called ‘‘contextuality:’’ An additional variable attached to a physical system is called ‘‘contextual’’ if ivalue depends not only on the physical quantity that itscribes, but also on the other physical quantities that canmeasured at the same time on the same system~in quantummechanics they correspond to commuting observables!. If,on the other hand, its value is completely independent othe other observables that the experimenter may decidmeasure at the same time, the additional variable is ca‘‘noncontextual;’’ one can then say that it describes a prerty of the physical system only, and not a combined prerty of the system and the measurement apparatus; ithave pre-existed in the system before any measurement.notion of distance is no longer relevant in this context;instance, the theorem applies to a single system with notension in space.

Let us first consider a spin 1 particle in quantum mechics, with three quantum statesu21&u0& and u11& as a basis ofa state space of the dimension three. The three componSx , Sy , andSz do not commute~they obey the usual commutation relation for the angular momentum!, but it is easyto show that all the squares of all these three operatorscommute; this is a specific property of angular momentumand can be seen by an elementary matrix calculation withusual operatorsS6 . Moreover, the sum of these squares isconstant~a c number! since

Sx21Sy

21Sz252\2. ~21!

It is not against any fundamental principle of quantum mchanics, therefore, to imagine a triple measurement ofobservablesSx

2, Sy2, and Sz

2; we know that the sum of thethree results will always be 2~from now on we drop thefactor\2, which plays no role in the discussion!. Needless tosay, the choice of the three orthogonal directions is copletely arbitrary, and the compatibility is ensured for achoice of this triad, but not more than one: The measuments for different choices remain totally incompatible.

In passing, we note that the measurement of the squarSx2

of one component cannot merely be seen as a measureof Sx followed by a squaring calculation made afterwardsthe experimentalist! Ignoring information is not equivalentnot measuring it~we come to this point in more detail, iterms of interferences and decoherence, at the end ofVI A !. There is indeed less information inSx

2 than in Sx

itself, since the former has only two eigenvalues~1 and 0!,while the latter has three~21 is also a possible result!. Whatis needed to measureSx

2 is, for instance, a modified SternGerlach system where the components of the wave funccorresponding to results61 are not separated, or where thare separated but subsequently grouped together in a


e

s

eder

-be

lltod--ayherx-

-

nts

do,e

-e

-

-

enty

ec.

n

ay

they makes them impossible to distinguish. Generally speing, in quantum mechanics, measuring the square of anerator is certainly not the same physical process as meaing the operator itself!

Now, suppose that we try to attach to each individual san EPR element of reality/additional variable that corsponds to the result of measurement ofSx

2; by symmetry, wewill do the same for the two other components, so that espin now gets three additional variablesl to which we mayattribute values that determine the possible results: 1 oThe results are described by functions of these variabwhich we denoteAx,y,z :

Ax50 or 1, Ay50 or 1, Az50 or 1. ~22!

At first sight, this seems to provide a total of eight possibties; but, if we want to preserve relation~21!, we have toselect among these eight possibilities only those threewhich two A’s are one, one is zero. As traditional, for thparticular spin we then attribute colors to the three orthonal directionsOx, Oy, andOz: The two directions that get anA51 are painted in red, the last in blue, Ref. 85.

The same operation can obviously be made for all posschoices of the triplet of directionsOxyz. A question whichthen naturally arises is: For an arbitrary direction, can oattribute a given color~a given value forAx! that remainsindependent of the context in which it was defined? Indewe did not define the value as a property of anOx directiononly, but in the context of two other directionsOy and Oz;the possibility of a context independent coloring is therefonot obvious. Can we for instance fixOzand rotateOx andOyaround it, and still keep the same color forOz? We are nowfacing an amusing little problem of geometry that we migcall ‘‘ternary coloring of all space directions.’’ Bell as weas Kochen and Specker showed that this is actually impsible; for a proof see either the original articles, or the exclent review~Ref. 6! given by Mermin.

In the same article, this author shows how the complitions of the geometrical problem may be entirely avoidedgoing to a space of states of dimension four instead of thHe considers two spin 1/2 particles and the following taof nine quantum variables~we use the same notation asSec. IV C!:

sx1 sx

2 sx1sx

2

sy2 sy

1 sy1sy

2

sx1sy

2 sy1sx

2 sz1sz

2. ~23!

All operators have eigenvalues61. It is easy to see why althree operators belonging to the same line, or to the sacolumn, always commute~the products of twos’s that anti-commute are commuting operators, since the commutaintroduces two21 signs, with canceling effects!. Moreover,the product of all three operators is always11, except thelast column for which it is21.30 Here, instead of an infinitenumber of triplet of directions in space, we have six grouof three operators, but the same question as above arCan we attribute a color to each of the nine elementsmatrix ~23!, red for result11 and yellow for result21, in away that is consistent with the results of quantum mechics? For this consistency to be satisfied, all lines and columshould either contain three red cases, or one red andyellow, except the last column, which will contain one

671F. Laloe

noba

steSnFoo

nlyibityero

telyulth

eapbo

neferao

nainich

i-seve

llyreadthim

thin

owe

-u

thkss

obare

ctionavem

nd-ed

areavenicst.’’thatnotthetother’sderara-maye totherily.es

ent,oon

,

wear-bi-eti-

aidin

wethe

on

beenno

gi-enebe-irstas-

orionsldheButing; inhis

ios:ay

three yellow cases~in order to correspond to21 instead of11!.

This little matrix coloring problem is much simpler thathe geometrical coloring problem mentioned above: It isviously impossible to find nine numbers with a product this at the same time equal to 1, condition on rows, and21,condition on columns~we note in passing that Mermin’reasoning is very close to that of Sec. IV C, which illustrahow similar the GHZ theorem and this form of the BKtheorem are!. Here, as in the three direction problem, nocontextuality leads us to an impossible coloring problem.another illustration of the impossibility, see also Sec. VIRef. 6 which deals with three 1/2 spins instead of two.

What can we conclude from this contradiction? Certaithat the predictions of quantum mechanics are incompatwith a noncontextual view on the EPR elements of realadditional variables, where the result of the measuremshould depend solely on the system measured—see fostance the discussion given in Ref. 86. But is this a goargument against these elements of reality, or at leasindication that, if they exist, their properties are completunexpected? Not really. As Bell noted in Ref. 3, ‘‘the resof an observation may reasonably depend not only onstate of the system~including hidden/additional variables!but also on the complete disposition of the apparatus.’’ This for instance no special conceptual difficulty in buildingtheory where additional variables are attributed to the apratuses, and where both kinds of additional variables collarate in order to determine the observed result. Violationsthe Bell theorem by quantum mechanics are therefore geally considered as much more significant quantum manitations than violations of the BKS theorem. For a genediscussion of the status of the various ‘‘impossibility therems’’ with emphasis on the BKS theorems, see Ref. 6.

V. NONLOCALITY AND ENTANGLEMENT:WHERE ARE WE NOW?

In view of the locality theorems as well as their violatioby the modern experimental results, which were not avable when the orthodox interpretation of quantum mechawas invented, some physicists conclude triumphantly: ‘‘Bowas right!,’’ while others will claim with the same enthusasm, ‘‘Bohr was wrong!’’ Both these opinions make sendepending on what aspect of the debate one wishes to faWe have already touched the question at the end of SIII B 3; here, we will just add that, whether one personafeels closer to the orthodox camp or to local realism, itmains clear that the line initiated by Einstein and Bell hthe decisive role in the last 50 years. In fact, they areones who pointed out the inadequate character of somepossibility theorems, as well as the crucial importance ofnotion of locality in all these discussions. This resultedmuch more progress and understanding than the simplestatement of the orthodox position. For instance, even nthe introduction of the reduction of the state vector is somtimes ‘‘explained’’ by invoking the ‘‘unavoidable perturbations that the measurement apparatus brings to the meassystem’’—see for instance the traditional discussion ofHeisenberg microscope which still appears in textbooBut, precisely, the EPR–Bell argument shows us that thionly a cheap explanation: In fact, the quantum descriptiona particle can be modified without any mechanical perturtion acting on it, provided the particle in question was p


-t

s

-r

f

le/ntin-dan

te

re

a-o-fr-

s-l

-

l-sr

,or.c.

-

e-

e

re-,-

rede!

isf-

-

viously correlated with another particle. So, a trivial effesuch as a classical recoil effect in a photon–electron colliscannot be the real explanation of the nature of the wpacket reduction! It is much more fundamentally quantuand may involve nonlocal effects.

Another lesson is that, even if quantum mechanics arelativity are not incompatible, they do not fit very well together: The notion of events in relativity, which are supposto be point-like in space–time, or the idea of causality,still basic notions, but not as universal as one could hthought before the Bell theorem. Indeed, quantum mechateaches us to take these notions ‘‘with a little grain of salStill another aspect is related to the incredible progressexperiments have made in the 20th century, whether orstimulated by fundamental quantum mechanics. One getsimpression that this progress is such that it will allow ushave access to objects at all kinds of scale, ranging frommacroscopic to the microscopic. Therefore, while at Bohtime one could argue that the precise definition of the borline between the macroscopic world of measurement apptuses was not crucial, or even academic, the questionbecome of real importance; it may, perhaps, even give risexperiments one day. All these changes, together, giveimpression that the final stage of the theory is not necessareached and that conceptual revolutions are still possible

In this section, we give a brief discussion of some issuthat are related to quantum nonlocality and entanglemwith some emphasis on those that are presently, or may sbe, the motivation of experiments~Sec. V B is an exceptionsince it is purely theoretical!. Going into details would cer-tainly bring us beyond the scope of this article, so thatwill limit ourselves to a few subjects, which we see as pticularly relevant, even if our choice may be somewhat artrary. Our main purpose is just to show that, even if theorcally it is really difficult to add anything to what thefounding fathers of quantum mechanics have already slong ago, it still remains possible to do interesting physicsthe field of fundamental quantum mechanics! Even iftreat the subject somewhat superficially, the hope is thatreader will be motivated to get more precise informatifrom the references.

A. Loopholes, conspiracies

One sometimes hears that the experiments that haveperformed so far are not perfectly convincing, and thatone should claim that local realism a` la Bell has been dis-proved. Strictly speaking, this is true: There are indeed local possibilities, traditionally called ‘‘loopholes,’’ which arstill open for those who wish to restore local realism. Ocan for instance deny the existence of any real conflicttween the experimental results and the Bell inequalities. Fof all, of course, one can always invoke trivial errors, suchvery unlikely statistical fluctuations, to explain why all experiments seem to ‘‘mimic’’ quantum mechanics so well; finstance, some authors have introduced ad hoc fluctuatof the background noise of photomultipliers, which woumagically correct the results in a way that would give timpression of exact agreement with quantum mechanics.the number and variety of Bell-type experiments supportquantum mechanics with excellent accuracy is now largeview of the results, very few physicists seem to take texplanation very seriously.

Then one could also think of more complicated scenarFor instance, some local unknown physical variables m

672F. Laloe

reerd

ega

athiertiaxts

asde

itliliteas

ha

f

alo

et-eino

ct

erthth

otthhehe

ans

alis

teeroc-rsat

airsde-foren ifs.

en-

88orsis-

.nt.edrs.eri-

thenly

r-the

ttere

ed,ent

e

re isted

h iseri-tedt-

. Onno

laceula-

ablents

ntsmeceon--rspin

fil-rio,1/2sachtheion

getion

couple together in a way that will give the~false! impressionof nonlocal results, while the mechanism behind themmains local. One possibility is that the polarization analyzmight, somehow, select a subclass of pairs which depentheir settings; then, for each choice~a, b!, only a small frac-tion of all emitted pairs would be detected; one could thassume that, when the orientation of the analyzers is chanby a few degrees, all the pairs that were detected beforeeliminated, and replaced with a completely different cegory of physical systems with arbitrary properties. In tsituation, everything becomes possible: One can ascribeach category of pairs whatever ad hoc physical propeare needed to reproduce any result, including those of qutum mechanics, while remaining in a perfectly local conte

Indeed, in the derivation of the Bell inequalities, one asumes the existence of ensemble averages over a nonbiwell-defined, ensemble of pairs, which are completely inpendent of the settingsa and b. Various proofs of the Bellinequalities are possible, but in many of them one explicwrites the averages with an integral containing a probabdistribution %(l); this function mathematically defines thensemble on which these averages are taken. The nonbiassumption is equivalent to assuming thatr is independent ofa andb; on the other hand, it is easy to convince oneself tthe proof of the Bell inequalities is no longer possible ifrbecomes a function ofa andb. In terms of the reasoning oSec. IV A 2, where no functionr was introduced, what wehave assumed is that the four numbersA, A8, B, andB8 areall attached to the same pair; ifM was built from more num-bers, such as numbers associated to different pairs, thebra would clearly no longer hold, and the rest of the proofthe inequality would immediately collapse.

Of course, no problem occurs if every emitted pair is dtected and provides two results61, one on each side, whaever the choice ofa andb ~and even if this choice is madafter the emission of the pair!. It then makes sense to obtathe ensemble average^M& from successive measurementsfour average valuesAB&, ^AB8&, etc. But, if many pairs areundetected, one cannot be completely sure that the deteefficiency remains independent of the settingsa andb; if it isnot, the four averages may in principle correspond to diffent subensembles, and there is no special reason whycombination by sum and difference should not exceedlimit of 2 given by the Bell theorem.31 The important point isnot necessarily to capture all pairs, since one could in theredefine the ensemble in relation with detection; but whaessential, for any perfectly convincing experiment onviolation of the Bell inequalities, is to make sure that tsample of counted events is completely independent of tsettingsa andb ~unbiased sample!. This, in practice, impliessome sort of selection~or detection! that is completely inde-pendent of the settings, which is certainly not the case inexperiment that detects only the particles that have crosthe analyzers.

An ideal situation would be provided by a device withtriggering button that could be used by an experimentawho could at will launch a pair of particles~with certainty!;if the pair in question was always analyzed and detecwith 100% efficiency, the loophole would definitely bclosed! When discussing thought experiments, Bell intduced in some of his talks the notion of ‘‘preliminary detetors’’ ~Ref. 87!, devices which he sketched as cylindethrough which any pair of particles would have to propagbefore reaching both ends of the experiment~where thea and


-son

nedre-sto

esn-.-ed,-

yy

ing

t

ge-f

-

f

ion

-eire

ryise

ir

yed

t,

d,

-

e

b dependent measurement apparatuses sit!; the idea was thatthe preliminary detectors should signal the presence of pand that, later, the corresponding pairs would always betected at both ends. The role of these cylinders was thereto make the definition of the sample perfectly precise, eveinitially the pairs were emitted by the source in all directionSuch classes of systems, which allow a definition of ansemble that is indeed totally independent ofa and b, aresometimes called ‘‘event ready detectors.’’ See also Ref.where Bell imagines a combination of veto and go detectassociated with the first detected particles in a ternary emsion, precisely for the purpose of better sample definition

Needless to say, in practice, the situation is very differeFirst, one should realize that, in all experiments performuntil now, most pairs are simply missed by the detectoThere are several reasons for this situation: In photon expments, the particles are emitted in all directions, whileanalyzers collect only a small solid angle and, therefore, oa tiny fraction of the pairs~this was especially true in theinitial experiments using photon cascades!; in more modernexperiments~Ref. 61!, the use of parametric photon convesion processes introduces a strong correlation betweendirection of propagation of the photons and a much becollection efficiency, but it still remains low. Moreover, thtransmission of the analyzers is less than one@it is actuallyless than 1/2 if ordinary photon polarization filters are usbut experiments have also been performed with birefringtwo-channel analyzers~Ref. 59!, which are not limited to50% efficiency#. Finally, the quantum efficiency of particldetectors~photomultipliers for photons! is not 100% either,so that pairs of particles are lost at the last stage too. Theno independent way to determine the sample of detecpairs, except of course the detection process itself, whicobviously a and b dependent; as a consequence, all expmental results become useful only if they are interprewithin a ‘‘no-biasing’’ assumption, considering that the seting of the analyzers does not bias the statistics of eventsthe other hand, we should also mention that there isknown reason why such a sample biasing should take pin the experiments, and that the possibility remains spective. For proposals of ‘‘loophole free’’ experiments,32 seeRefs. 89 and 90; actually, there now seems to be a reasonhope that this loophole will be closed by the experimewithin the next few years.

Other loopholes are also possible: Even if experimewere done with 100% efficiency, one could also invoke sopossibilities for local processes to artificially reproduquantum mechanics. One of them is usually called the ‘‘cspiracy of the polarizers’’~actually, ‘‘conspiracy of the analyzers’’ would be more appropriate; the word polarizer refeto the experiments performed with photons, where the sorientation of the particles is measured with polarizingters; but there is nothing specific of photons in the scenawhich can easily be transposed to massive spinparticles!—or also ‘‘communication loophole.’’ The idea ithe following: Assume that, by some unknown process, eanalyzer could become sensitive to the orientation ofother analyzer; it would then acquire a response functwhich depends on the other setting and the functionA couldacquire a dependence on botha andb. The only way to beatthis process would be to choose the settingsa and b at thevery last moment, and to build an experiment with a lardistance between the two analyzers so that no informacan propagate~at the speed of light! between the two of

673F. Laloe

nsa-so

a-

inr,

luethO

rlaot

th-or

hithicinmlof

era

mdd...nriem

or

legundc

ghacfi-icntefc

ossr

osri

thre

mm.

inglity.an-n-to

tly

aretail

he

eri-nedua-esry

ryhtheas asolizeonet all

of, at

tionnees-ainthed a

x-e-l

elln-to

hers,it isnn

riestlyrdduc-ri-

ondif

ocal

them. A first step in this direction was done by Aspect aco-workers in 1982~Ref. 91!, but more recent experimenthave beautifully succeeded in excluding this possibility inespecially convincing way~Ref. 92!. So there no longer exists a real conspiracy loophole; quantum mechanics seemstill work well under these more severe time dependent cditions.

Along a similar line is what is sometimes called the ‘‘ftalistic loophole’’ ~or also ‘‘superdeterminism’’!. The idea isto put into question an implicit assumption of the reasonthat leads to the Bell theorem: the completely arbitrachoice of the settingsa andb by the experimenters. Usuallya andb are indeed considered as free variables: their vathat are not the consequence of any preliminary eventtook place in the past, but those of a free human choice.the other hand, it is true that there is always some ovebetween the past cones of two events, in this case the chof the settings. It is therefore always possible in theoryassume that they have a common cause;a andb are then nolonger free parameters, but variables that can fluctuate~inparticular, if this cause itself fluctuates! with all kinds ofcorrelations. In this case, it is easy to see that the proof ofBell theorem is no longer possible,33 so that any contradiction with locality is avoided. What is then denied is the ntion of free will of the experimenters, whose decisions aactually predetermined, without them being aware of tfact; expressed more technically, one excludes fromtheory the notion of arbitrary external parameters, whusually define the experimental conditions. This price bepaid, one could in theory build an interpretation of quantumechanics that would remain at the same time realistic,cal, and~super! deterministic, and would include a sort ophysical theory of human decision. This is, of course, a vunusual point of view, and the notion of arbitrary externparameters is generally accepted; in the words of Bell~Ref.93!: ‘‘A respectable class of theories, including quantutheory as it is practised, have free external variables in ation to those internal to and conditioned by the theoryThey are invoked to represent the experimental conditioThey also provide a point of leverage for free willed expementers... .’’ Needless to say, the fatalist attitude in scienceven more subject to the difficulties of orthodox quantumechanics concerning the impossibility to develop a thewithout observers, etc.

We could not honestly conclude this section on loophowithout mentioning that, while most specialists acknowledtheir existence, they do not take them too seriously becaof their ‘‘ad hoc’’ character. Indeed, one should keep in mithat the explanations in question remain artificial, inasmuthey do not rest on any precise theory: No-one has the sliest idea of the physical processes involved in the conspiror of how pair selection would occur in a way that is sufciently complex to perfectly reproduce quantum mechanBy what kind of mysterious process would experimemimic quantum mechanics so perfectly at low collectionficiencies, and cease to do so at some threshold of efficienBell himself was probably the one who should have mliked to see that his inequalities could indeed be used alogical tool to find the limits of quantum mechanics; nevetheless, he found these explanations too unaesthetic treally plausible. But in any case logic remains logic: Yethere still remains a slight possibility that, when the expements reach a level of efficiency in pair collection whereloophole becomes closed, the results concerning the cor


d

n

ton-

gy

satnpiceo

e

-esehg

-

yl

i-.s.-is

y

sese

ht-y,

s.s-y?ta

-be,-ela-

tion rates will progressively deviate from those of quantumechanics to reach values compatible with local realisWho knows?

B. Locality, contrafactuality

One can find in the literature various attitudes concernthe exact relation between quantum mechanics and locaSome authors consider that the nonlocal character of qutum mechanics is a well-known fact, while for others quatum nonlocality is an artifact created by the introduction inquantum mechanics of notions which are foreign to it~typi-cally the EPR elements of reality!. Lively discussions to de-cide whether or not quantum mechanics in itself is inherennonlocal have taken place and are still active~Refs. 94–96!;see also Refs. 21, 97, and 98. Delicate problems of logicinvolved and we will not discuss the question in more dehere.

What is easier to grasp for the majority of physicists is tnotion of ‘‘contrafactuality’’ ~Ref. 95!. A counterfactual rea-soning consists in introducing the results of possible expments that can be envisaged for the future as well-defiquantities, and valid mathematical functions to use in eqtions, even if they are still unknown—in algebra one writunknown quantities in equations all the time. This is venatural: As remarked by d’Espagnat~Refs. 99 and 100! andby Griffiths ~Ref. 101!, ‘‘counterfactuals seem a necessapart of any realistic version of quantum theory in whicproperties of microscopic systems are not created bymeasurements.’’ One can also see the EPR reasoningjustification of the existence of counterfactuals. But it alremains true that, in practice, it is never possible to reamore than one of these experiments: For a given pair,has to choose a single orientation of the analyzers, so thaother orientations will remain forever in the domainspeculation. For instance, in the reasoning of Sec. IV A 2least some of the numbersA, A8, B, andB8 are counterfac-tuals, and we saw that using them led us to a contradicwith quantum mechanics through the Bell inequalities. Ocould conclude that contrafactuality should be put into qution in quantum mechanics; alternatively, one could maintcounterfactual reasoning, but then the price to pay isexplicit appearance of nonlocality. We have already quotesentence by Peres~Ref. 53! which wonderfully summarizesthe situation as seen within orthodoxy: ‘‘unperformed eperiments have no results;’’ as Bell once regretfully rmarked~Ref. 93!: ‘‘It is a great inconvenience that the reaworld is given to us once only!’’

But, after all, one can also accept contrafactuality as was explicit nonlocality together, and obtain a perfectly cosistent point of view; it would be a real misunderstandingconsider the Bell theorem as an impossibility theorem, eitfor contrafactuality, or for hidden variables. In other wordand despite the fact that the idea is still often expressed,not true that the Bell theorem is a new sort of Von Neumatheorem. The reason is simple: Why require that theowith contrafactuality/additional variables should be explicilocal at all stages, while it is not required from standaquantum mechanics? Indeed, neither the wave packet retion postulate, nor the calculation of correlation of expemental results in the correlation point of view~Sec. VI A!,nor again the expression of the state vector itself, correspto mathematically local calculations. In other words, evenone can discuss whether or not quantum mechanics is l

674F. Laloe

itsesllyaisw-elo—

it-

me

t ws

er

t

li

gerg

axuiad

tat

ri-

n–

th

sth

r-at

pstre

f a, inthe

btle

t,’’ne

en-

ncelas-

ache

nicsv-

ea-

om

itro-

ofaatic.re-e

hatlya-n-

r-thatal

ure-ases

orese

veduch

withag-

ods

or not at a fundamental level, it is perfectly clear thatformalism is not; it would therefore be just absurd to requa local formalism from a nonorthodox theory—especiawhen the theory in question is built in order to reproduceresults of quantum mechanics! As an illustration of thpoint, as seen from theories with additional variables,quote Goldstein~Ref. 16!: ‘‘in recent years it has been common to find physicists... failing to appreciate that what Bdemonstrated with his theorem was not the impossibilityBohmian mechanics, but rather a more radical implicationnamely non-locality—that is intrinsic to quantum theoryself.’’

C. ‘‘All-or-nothing coherent states;’’ decoherence

In this section, we first introduce many particle quantustates which have particularly remarkable correlation propties; then we discuss more precisely a phenomenon thahave already introduced above, decoherence, which tendreduce their lifetime very efficiently, especially if the numbof correlated particles is large.

1. Definition and properties of the states

The states that we will call ‘‘all-or-nothing coherenstates’’ ~or all-or-nothing states for short! could also becalled ‘‘many-particle GHZ states’’ since they are generazations of~16! to an arbitrary numberN of particles:

uC&5au1:1;2:1;....;N:1&1bu1:2;2:2;...;N:2&,~24!

where a and b are arbitrary complex numbers satisfyinuau21ubu251. In fact, the most interesting situations genally occur whena andb have comparable modulus, meaninthat there are comparable probabilities to find the systemstates where all, or none, of the spins is flipped~hence thename we use for these states!; whena andb are both equalto 1/&, these states are sometimes called ‘‘states of mmum entanglement’’ in the literature, but since the measof entanglement for more than two particles is not triv~several different definitions have actually been proposethe literature!, here we will use the words ‘‘all-or-nothingstates’’ even in this case.

In order to avoid a frequent cause of confusion, andbetter emphasize the peculiarities of these entangled stlet us first describe a preparation procedure that wouldnotlead to such a state. Suppose thatN spin 1/2 particles ori-ented along directionOx enter a Stern–Gerlach magnet oented along directionOz, or more generally thatN particlescross a filter~polarization dependent beam splitter, SterGerlach analyzer, etc.! while they are initially in a statewhich is a coherent superposition of the eigenstates offilter, with coefficientsa1 andb1 . The effect of the filter onthe group of particles is to put them into a state which iproduct of coherent superpositions of the two outputs offilter, namely:

uC&5@au1:1&1bu1:2&] ^ @au2:1&1bu2:2&]

^ ...^ @auN:1&1buN:2&]. ~25!

The point we wish to make is that this state is totally diffeent from~24!, since it contains many components of the stvector where some of the spins are up, some down. In~25!,each particle is in a coherent superposition of the two sstates, a situation somewhat analogous to a Bose–Eincondensate where all particles are in the same cohe


t

ll

e

lf

r-eto

-

-

in

i-relin

oes,

is

ae

e

ineinnt

state—for instance two states located on either sides opotential barrier as in the Josephson effect. By contrast~24! all spins, or none, are flipped from one component toother34 so that the coherence is essentially anN-body coher-ence only; it involves entanglement and is much more suthan in ~25!. For instance, one can show~Ref. 76! that thecoherence in question appears only ‘‘at the last momenwhen all particles are taken into account: As long as oconsiders any subsystem of particles, evenN21, it exhibitsno special property and the spins are correlated in an elemtary way~as they would be in a classical magnet!; it is onlywhen the last particle is included that quantum cohereappears and introduces effects which are completely noncsical.

Assume for instance that

a51/&, b5eiw/& ~26!

and that a measurement is performed of a component of espin that belongs to theOxyplane and is defined by its anglu1 with Ox for the first particle,u2 for the second,...uN forthe last. It is an easy exercise of standard quantum mechato show that the product of all results has the following aerage value:

E~u1 ,u2 ,...,uN!5cos~u11u21¯ .uN1w! ~27!

~we take the usual convention where the results of each msurement is61!. For instance, each time the sumu11u2

1¯uN1w is equal to an integer even multiple ofp, theaverage is 1, indicating that the result is certain and free frany fluctuation~in the same way, an odd multiple ofp wouldgive a certain value,21!. Indeed, the result of the quantumcalculation may look extremely simple and trivial; butturns out that, once more, it is totally impossible to repduce the oscillations contained in~27! within local realism.In the caseN52, this is of course merely the consequencethe usual Bell theorem; as soon asN becomes 3 or takeslarger value, the contradiction becomes even more dramActually, if one assumes that a local probabilistic theoryproduces~27! only for some sets of particular value of thanglesu ~those for which the result is certain!, one can show~Ref. 76! that the theory in question necessarily predicts tE is independent of allu’s. The average keeps a perfectconstant value11! Indeed, the very existence of the oscilltion predicted by~27! can be seen as a purely quantum nolocal effect~as soon asN>2!.

This is, by far, not the only remarkable property of all-onothing coherent states. For instance, it can be shownthey lead to exponential violations of the limits put by locrealistic theories;~Ref. 83!; it has also been pointed out~Ref.102! that these states, when relation~26! is fulfilled ~they arethen called ‘‘maximally correlated states’’ in Ref. 102!, haveinteresting properties in terms of spectroscopic measments: The frequency uncertainty of measurements decreas 1/N for a given measurement time, and not as 1/AN as anaive reasoning would suggest. This is of course a mfavorable situation, and the quantum correlation of thestates may turn out to be, one day, the source of improaccuracy on frequency measurements. How to create sstates with massive particles such as atoms, and notphotons as usual, was demonstrated experimentally by Hley et al. in 1997 ~Ref. 103! in the caseN52. We havealready mentioned in Sec. IV C that entanglement withN53 was reported in Refs. 81 and 82. Proposals for meth

675F. Laloe

stiafu

r

tiaannn

aporga

a

hilen

m

too

omss

nm

fer

eit

arth

cmglreccv

hiofeic

n as

ore

e

uct

oton

ro

ceel-lose

in-re-te

lecat-

tialibleternotentthey

ind innceeomen’’er-notde-

toat-cketady

theackper-tionldn; weat athis

to generalize to larger values ofN with ions in a trap wereput forward by Molmeret al. ~Ref. 104!; the idea exploitsthe motion dependence of resonance frequencies for a syof several ions in the same trap, as well as on some partdestructive interference effects. The scheme was successput into practice in a very recent experiment by Sackettet al.~Ref. 105! where ‘‘all-or-nothing states’’ were created foN52 as well asN54 ions in a trap.

2. Decoherence

We have defined in Sec. II A decoherence as the inipart of the phenomenon associated with the Von Neuminfinite regress: Coherent superpositions tend to constapropagate toward the environment, they involve more amore complex correlations with it, so that they become ridly completely impossible to detect in practice. To see mprecisely how this happens, let us for instance consider astate~24!; we now assume that the single particle statesu1&and u2& are convenient notations for two states where a pticle has different locations in space~instead of referringonly to opposite spin directions!: This will happen for in-stance if the particles cross a Stern–Gerlach analyzer wcorrelates the spin directions to the positions of the particUnder these conditions, it is easy to see that the coherecontained in the state vector becomes extremely fragileany interaction with environment. To see why, let us assuthat an elementary particle~photon for instance!, initially instateuk0&, interacts with the particles. It will then scatter ina quantum state that is completely different, dependingwhere the scattering event took place: If the scattering atare in the first stateu1&, the photon is scattered by the atominto stateuk1&; on the other hand, if it interacts with atomin stateu2&, it is scattered into stateuk2&.35 As soon as thenew particle becomes correlated with the atoms, the ostate vector that can be used to describe the systemincorporate this new particle as well, and becomes

uC8&5au1:1;2:1;....;N:1& ^ uk1&

1bu1:2;2:2;...N:2& ^ uk2&. ~28!

Assume now that we are interested only in the system oNatoms; the reason might be, for instance, that the scattphoton is impossible~or very difficult! to detect~e.g., it maybe a far-infrared photon!. It is then useful to calculate thpartial trace over this photon in order to obtain the densoperator which describes the atoms only. A straightforwcalculation shows that this partial trace can be written, inbasis of the two statesu1,1,1,..& and u2,2,2,..&:

r5S uau2 ab* ^k2uk1&

a* b^k1uk2& ubu2 D ~29!

~for the sake of simplicity we assume that the statesuk6& arenormalized!. We see in this formula that, if the scalar produ^k2uk1& was equal to one, the density matrix of the atowould not be affected at all by the scattering of the sinphoton. But this would assume that the photon is scatteexactly into the same state, independent of the spatial lotion of the scatterers! This cannot be true if the distanbetween the locations is much larger than the photon walength. Actually, it is much more realistic to assume that tscalar product is close to zero, which means that thediagonal element of~29!, in turn, becomes almost zero. Wthen conclude that the scattering of even a single part


emllylly

ln

tlyd-ein

r-

chs.cetoe

ns

lyust

ed

yde

tseda-ee-sf-

le

destroys the coherence between atomic states, as soothey are located at different places.

The situation becomes even worse when more and mphotons~assumed to be all in the same initial stateuk0&! arescattered, since one then has to replace~28! by the state:

uC8&5au1:1;2:1;....;N:1& ^ uk1&uk18 &uk19 &...

1bu1:2;2:2;...N:2& ^ uk2&uk28 &uk29 ... ~30!

with obvious notation~the states withn primes correspond tothe n21th scattered photon!; the same calculation as abovthen provides the following value forr:

S uau2 ab* ^k2uk1&^k28 uk18 &...

a* b^k1uk2&^k18 uk28 &... ubu2 D .

~31!

Since we now have, in the off-diagonal elements, the prodof all single scalar productk2uk1&, it is clear that theseelements are even more negligible than when a single phis scattered. Actually, as soon as the two statesuk1& anduk2& are not strictly identical, they tend exponentially to zewith the number of scattering events.

This is a completely general property: Objects~especiallymacroscopic objects! have a strong tendency to leave a train the environment by correlating themselves with anyementary particle which passes by; in the process, theytheir own coherence, which regresses into a coherencevolving the environment and more and more complex corlations with it ~the scattered photon, in turn, may correlawith other particles!; soon it becomes practically impossibto detect. The phenomenon is unavoidable, unless the stering properties of both states symbolized byu1& and u2&are exactly the same, which excludes any significant spaseparation between the states. In particular, it is impossto imagine that a cat, whether dead or alive, will scatphotons exactly in the same way, otherwise we couldeven see the difference! This shows how fragile cohersuperpositions of macroscopic objects are, as soon asinvolve states that can be seen as distinct.

We are now in a position where we can come backmore detail to some questions that we already discussepassing in this text, and which are related to decohereand/or the Schro¨dinger cat. The first question relates to thconceptual status of the phenomenon of decoherence. Sauthors invoke this phenomenon as a kind of ‘‘explanatioof the postulate of wave packet reduction: When the supposition of the initial system becomes incoherent, are wein presence of a statistical mixture that resembles thescription of a classical object with well-defined~but ignored!properties? On this point, we do not have much to addwhat was already said in Sec. II: This explanation is unsisfactory because the purpose of the postulate of wave pareduction is not to explain decoherence, which can alrebe understood from the Schro¨dinger equation, but theuniqueness of the result of the measurement—in fact,effect of the wave packet reduction is sometimes to put bthe measured subsystem into a pure state, which is thefect opposite of a statistical mixture, so that the real quesis to understand how the~re!emergence of a pure state shoube possible~Ref. 132!. Indeed, in common life, as well as ilaboratories, one never observes superposition of resultsobserve that Nature seems to operate in such a way thsingle result always emerges from a single experiment;

676F. Laloe

to.thhralthiereo

at

eaiti

caituhe

str

est

et

ico

ic-

ith

nao

Bug

tb

nta

miste

omo

rochts

art

natn

met thatalg ofrencelf; itde-ixl-

hatrta-

s

R

theordsedstep,

enge;s-

es-e

key.og-eepcureich

un-ak:besic

bycor-dom

mu-ns,theand

justngeheeenthe

g.y

isnto

ionteto

will never be explained by the Schro¨dinger equation, sinceall that it can do is to endlessly extend its ramifications inthe environment, without ever selecting one of them only

Another way to say the same thing is to emphasizelogical structure of the question. The starting point is tnecessity for some kind of limit of the validity of the lineaSchrodinger equation, the initial reason being that a lineequation can never predict the emergence of a single resuan experiment. The difficulty is where and how to create tborder. Logically, it is then clear that this problem will nevbe solved by invoking any process that is entirely containin the linear Schro¨dinger equation, such as decoherenceany other similar linear process; common sense says thone stays in the middle of a country one never reachesborders. Actually, no one seriously doubts that a typical msurement process will involve decoherence at some instage, but the real question is what happens after.

Pressed to this point, some physicists reply that onealways assume that, at some later stage, the superposresolves into one of its branches only; this is of course trbut this would amount to first throwing a problem out by tdoor, and then letting it come back through the window!~Seethe discussions above, for instance, on the status of thevector and the necessity to resolve the Wigner friend padox.! A more logical attitude, which is indeed sometimproposed as a solution to the problem, is to consider thatnatural complement of decoherence is the Everett interprtion of quantum mechanics~see Sec. VI E!; indeed, this pro-vides a consistent interpretation of quantum mechanwhere the emergence of a single result does not have texplained, since it is assumed never to take place~the Schro¨-dinger equation then has no limit of validity!. But, of course,in this point of view, one has do deal with all the intrinsdifficulties of the Everett interpretation, which we will discuss later.

Concerning terminology, we have already mentionedSec. II C that, during the last few years, it has become rafrequent to read the words ‘‘Schro¨dinger cat’’ ~SC! used inthe context of states such as~24! for small values ofN ~ac-tually even for a single ion, whenN51!. This is a redefini-tion of the words, since the essential property of the origicat was to have a macroscopic number of degree of freedwhich is not the case for a few atoms, ions, or photons.let us now assume that someone succeeded in preparinall-or-nothing state with a very large value ofN. Would thatbe a much better realization of the Schro¨dinger cat as meanby its inventor? To some extent, yes, since the cat canseen as a symbol of a system of many particles that chatheir state, when one goes from one component of the svector to the other. Indeed, it is likely that many of the atoof a cat take part in different chemical bonds if the catalive or dead, which puts them in a different quantum staBut it seems rather hard to invent a reason why every atevery degree of freedom, should necessarily be in anthogonal state in each case, while this is the essential perty of ‘‘all-or-nothing states.’’ In a sense they do too mufor realizing a standard Schro¨dinger cat, and the concepremain somewhat different, even for large values ofN.

From an experimental point of view, decoherence isinteresting physical phenomenon that is certainly wostudying in itself, as recent experiments have illustrated~Ref.36!; a result of these studies and of the related calculatioamong others, is to specify the basis in the space of stthat is relevant to the decoherence process, as a functio


ee

rins

drif

its-

al

nione,

atea-

hea-

s,be

ner

lm,tan

egete

s

.,

r-p-

nh

s,esof

the coupling Hamiltonian, as well as the characteristic ticonstants that are associated. One can reasonably expecmore experiments on decoherence will follow this initibreakthrough and provide a more detailed understandinmany aspects of the phenomenon. Nevertheless, decoheis not to be confused with the measurement process itseis just the process which takes place just before: Duringcoherence, the off-diagonal elements of the density matr36

vanish ~decoherence!, while in a second step all diagonaelements but one should vanish~emergence of a single result!.

D. Quantum cryptography, teleportation

In Secs. V D 1 and V D 2, we discuss two concepts thave some similarity, quantum cryptography and telepotion.

1. Sharing cryptographic keys by quantum measurement

A subject that is relatively new and related to the EPcorrelations is quantum cryptography~Refs. 106 and 107!.The basic idea is to design a perfectly safe system fortransmission at a distance of a cryptographic key—the wrefers to a random series of numbers 0 or 1, which are uto code, and subsequently decode, a message. In a firstthe two remote correspondents A~traditionally called Alice!and B~traditionally called Bob! share this key; they then usit to code, or to decode, all the messages that they exchaif the key is perfectly random, it becomes completely imposible for anyone who does not know it to decode any msage, even if it is sent publicly. But if, during the initial stagof key exchange, someone can eavesdrop the message~theactor in question is traditionally called Eve!, in the futurehe/she will be able to decode all messages sent with thisExchanging keys is therefore a really crucial step in cryptraphy. The usual strategy is to take classical methods to kthe secret: storage in a safe, transport of the keys by semeans, etc.; it is always difficult to assess its safety, whdepends on many human factors.

On the other hand, quantum sharing of keys relies on fdamental physical laws, which are impossible to breHowever clever and inventive spies may be, they will notable to violate the laws of quantum mechanics! The baidea is that Alice and Bob will create their common keymaking quantum measurements on particles in an EPRrelated state; in this way they can generate series of rannumbers that can be subsequently used as a secret comnication. What happens if Eve tries to intercept the photofor example by coupling some elaborate optical device tooptical fiber where the photons propagate between AliceBob, and then making measurements?

If she wants to operate unnoticed, she clearly cannotabsorb the photons in the measurement; this would chathe correlation properties observed by Alice and Bob. Tnext idea is to try to ‘‘clone’’ photons, in order to makseveral identical copies of the initial photon; she could thuse a few of them to measure their properties, and resendlast of them on the line so that no-one will notice anythinBut, it turns out that ‘‘quantum cloning’’ is fundamentallimpossible: Within the rules of quantum mechanics, thereabsolutely no way in which several particles can be put ithe same arbitrary and unknown stateuw& as one given par-ticle, Refs. 108 and 109—see also Ref. 110 for a discussof multiple cloning. In Appendix D we discuss why, if stacloning were possible, it would be possible to apply it

677F. Laloe

thm

asy.iom

tonobonprahos

agic

hi-

toto

t

tany

inicest nab12ticto

lloo

mo, tth

bndtio

te-

wng

elftothe

temyec-s a

nlyta-ns-m-ryex-er-mram-in-

bi-n!nd,ect

esrio.Al-his,lachinrta--

hisatom-heveer-le-and

the

uffi-ralthef theananedem;

achtinu-ateke

sthe

atent

’’omin

each particle of an EPR pair of correlated particles; thenmultiple realization of the states could be used to transinformation on the orientationsa andb used by the experi-menters. Since such a scheme would not be subject tominimum time delay, it could also transmit messages atperluminal velocities, and be in contradiction with relativitFortunately for the consistency of physics, the contradictis avoided by the fact that cloning of single systems is ipossible in quantum mechanics!

So, whatever Eve does to get some information will aumatically change the properties of the photons at both eof the optical fiber, and thus be detectable by Alice and Bif they carefully compare their data and their correlatirates. Of course, they do not necessarily have a way tovent Eve’s eavesdropping, but at least they know what dcan be used as a perfectly safe key. There is actually a wvariety of schemes for quantum cryptography, some baon the use of EPR correlated particles, others not~Ref. 107!;but this subject is beyond our scope here.

2. Teleporting a quantum state

The notion of quantum teleportation~Ref. 111! is alsorelated to quantum nonlocality; the idea is to take advantof the correlations between two entangled particles, whinitially are for instance in state~24! ~for N52!, in order toreproduce at a distance any arbitrary spin state of a tparticle. The scenario is the following: Initially, two entangled particles propagate toward two remote regionsspace; one of them reaches the laboratory of the first acAlice, while the second reaches that of the second acBob; a third particle in quantum stateuw& is then provided toAlice in her laboratory; the final purpose of the scenario isput Bob’s particle into exactly the same stateuw&, whatever itis ~without, of course, transporting the particle itself!. Onethen says that stateuw& has been teleported.

More precisely, what procedure is followed in teleportion? Alice has to resist the temptation of performing ameasurement on the particle in stateuw& to be teleported;instead, she performs a ‘‘combined measurement’’ thatvolves at the same time this particle as well as her partfrom the entangled pair. In fact, for the teleportation procto work, an essential feature of this measurement is thadistinction between the two particles involved must be estlished. With photons one may for instance, as in Ref. 1direct the particles onto opposite sides of the same opbeam splitter, and measure on each side how many phoare either reflected or transmitted; this device does not aone to decide from which initial direction the detected phtons came, so that the condition is fulfilled. Then, Alice comunicates the result of the measurement to Bob; this is dby some classical method such as telephone, e-mail, etc.is by a method that is not instantaneous but submitted tolimitations related to the finite velocity of light. Finally, Boapplies to his particle a unitary transformation that depeon the classical information he has received; this operaputs it exactly into the same stateuw& as the initial state of thethird particle, and realizes the ‘‘teleportation’’ of the staThe whole scenario is ‘‘mixed’’ because it involves a combination of transmission of quantum information~throughthe entangled state! and classical information~the phone callfrom Alice to Bob!.

Teleportation may look trivial, or magical, depending hoone looks at it. Trivial because the possibility of reproduci


eit

nyu-

n-

-ds,

e-tale

ed

eh

rd

ofr,r,

o

-

-leso-,alnsw--nehate

sn

.

at a distance a state from classical information is not in itsa big surprise. Suppose for instance that Alice decidedchoose what the teleported state should be, and filteredspin ~she sends particles through a Stern–Gerlach sysuntil she gets a11 result37!; she could then ask Bob btelephone to align his Stern–Gerlach filter in the same dirtion, and to repeat the experiment until he also observe11 result. This might be called a trivial scenario, based oon the transmission of classical information. But teleportion does much more than this! First, the state that is traported is not necessarily chosen by Alice, but can be copletely arbitrary. Second, a message with only binaclassical information, such as the result of the combinedperiment made by Alice in the teleportation scheme, is ctainly not sufficient information to reconstruct a quantustate; in fact a quantum state depends on continuous paeters, while results of experiments correspond to discreteformation only. Somehow, in the teleportation process,nary information has turned into continuous informatioThe latter, in classical information theory, would correspoto an infinite number of bits~in the trivial scenario abovesending the complete information on the state with perfaccuracy would require an infinite time!.

Let us come back in more detail to these two differencbetween teleportation and what we called the trivial scenaConcerning the arbitrary character of the state, of courseice may also, if she wishes, teleport a known state. For tbeforehand, she could for instance perform a Stern–Gerexperiment on the third particle in order to filter its spstate. The remarkable point, nevertheless, is that telepotion works exactly as well if she is given a spin in a completely unknown state, by a third partner for instance; in tcase, it would be totally impossible for her to know whquantum state has been sent just from the result of the cbined experiment. A natural question then arises: If sknows nothing about the state, is it not possible to improthe transmission efficiency by asking her to try and detmine the state in a first step, making some trivial singparticle measurement? The answer to the question is no,for a very general reason: It is impossible to determineunknown quantum state of a single particle~even if one ac-cepts only ana posterioridetermination of a perturbed state!;one quantum measurement clearly does not provide scient information to reconstruct the whole state; but sevemeasurements do not provide more information, sincefirst measurement has already changed the spin state oparticle. In fact, acquiring the complete information onunknown spin-1/2 state would require Alice to start frominfinite number of particles that have been initially preparinto this same state, and to perform measurements on ththis is, once more, because the information given by emeasurement is discrete while the quantum state has conous complex coefficients. Alice cannot either clone the stof the single particle that she is given, in order to maseveral copies of it and measure them~see the precedingsection and Appendix D!. So, whatever attempt Alice maketo determine the state before teleportation will not helpprocess.

Concerning the amount of transmitted information, whBob receives has two components: classical information sby Alice, with a content that is completely ‘‘uncontrolled,since it is not decided by her, but just describes the randresult of an experiment; quantum information containedthe teleported state itself~what we will call a ‘‘q bit’’ in Sec.

678F. Laloe

he

tehalleret

uma

froul

sveosen-heisa

-o

elebts

leeeubenngpls

ts

hhwanhtah

ch

tw

otete

re

-

ofh antsallmrmm

heh-s-ed,ofby

en-the

l in

las-er-

n bem

bes,thisis

re-t

u-pic

g theea-

for-s-

nf a

ion,efultumof

tlyive

re-be

ap-de-

iningces

s

V E! and can possible be controlled. We know that neitBob nor Alice can determine the teleported state fromsingle occurrence, but also that Alice can prepare the stabe teleported by a spin filtering operation in a direction tshe decides arbitrarily; Bob then receives some controinformation as well. For instance, if the teleportation ispeated many times, by successive measurements on theported particles, Bob will be able to determine its quantstate with arbitrary accuracy, including the direction that wchosen by Alice; he therefore receives a real messageher~for a discussion of the optimum strategy that Bob shoapply, see Ref. 113!.

If one wishes to describe teleportation things in a sentional way, one could explain that, even before Bob receiany classical information, he has already received ‘‘almall the information’’ on the quantum state, in fact all thcontrollable information since the classical message doeshave this property; this ‘‘information’’ has come to him instantaneously, exactly at the time when Alice performedcombined experiment, without any minimum delay thatproportional to the distance covered. The rest of the informtion, which is the ‘‘difference’’ between a continuous ‘‘information’’ and a discrete one, comes only later and is,course, subject to the minimum delay associated with rtivity. But this is based on an intuitive notion of ‘‘differencbetween quantum/controllable and classical/noncontrollainformation’’ that we have not precisely defined; needlesssay, this should not be taken as a real violation of the baprinciples of relativity!

Finally, has something really been transported in the teportation scheme, or just information? Not everyone agron the answer to this question, but this may be just a discsion on words, so that we will not dwell further on the suject. What is perfectly clear in any case is that the essencthe teleportation process is completely different from ascenario of classical communication between human beiThe relation between quantum teleportation and Bell-tynonlocality experiments is discussed in Ref. 114; see aRef. 115 as well as Ref. 116 for a review of recent resul

E. Quantum computing and information

Another recent development is quantum computing~Refs.117–119!. Since this is still a rather new field of researcour purpose here cannot be to give a general overview, wnew results are constantly appearing in the literature. Wetherefore slightly change our style in the present section,limit ourselves to an introduction of the major ideas, at tlevel of general quantum mechanics and without any dewe will provide references for the interested reader wwishes to learn more.

The general idea of quantum computing~Ref. 120! is tobase numerical calculations, not on classical ‘‘bits,’’ whican be only in two discrete states~corresponding to 0 and 1in usual binary notation!, but on quantum bits, or ‘‘q bits,’’that is on quantum systems that have access to adimensional space of states; this means thatq bits cannotonly be in one of the two statesu0& and u1&, but also in anylinear superposition of them. It is clear that a continuumstates is a much ‘‘larger’’ ensemble than two discrete staonly; in fact, for classical bits, the dimension of the staspace increases linearly with the number of bits~for instance,the state of three classical bits defines a vector with thcomponents, each equal to 0 or 1!; for q bits, the dimensionincreases exponentially~this is a property of the tensor prod


ratotd

-ele-

sm

d

a-st

ot

r

-

fa-

leoic

-s

s--ofys.eo

.

,ileilld

eil;o

o-

fs

e

uct of spaces; for instance, for threeq bits the dimension ofspace is 2358!. If one assumes that a significant numberq bits is available, one gets access to a space state witenormous ‘‘size,’’ where many kinds of interference effeccan take place. Now, if one could somehow makebranches of the state vector ‘‘work in parallel’’ to perforindependent calculations, it is clear that one could perfomuch faster calculations, at least in theory. This ‘‘quantuparallelism’’ opens up many possibilities; for instance, tnotion of unique computational complexity of a given matematical problem, which puts limits on the efficiency of clasical computers, no longer applies in the same way. Indeit has been pointed out in Ref. 121 that the factorizationlarge numbers into prime factors may become faster thanclassical methods, and by enormous factors. Similarhancements of the speed of computation are expected insimulation of many-particle quantum systems~Ref. 122!. Forsome other problems the gain in speed is only polynomiatheory, still for some others there is no gain at all.

Fundamentally, there are many differences between csical and quantum bits. While classical bits have two refence states that are fixed once and for all,q bits can use anyorthogonal basis in their space of states. Classical bits cacopied at will and ad infinitum, while the no-cloning theorementioned in the preceding section~see also Appendix D!applies toq bits. On the other hand, classical bits cantransmitted only into the forward direction of light conewhile the use of entanglement and teleportation removeslimitation for q bits. But we have to remember that theremore distance between quantumq bits and information thanthere is for their classical bits: In order to transmit andceive usable information fromq bits, one has to specify whakind of measurements are made with them~this is related tothe flexibility on space state basis mentioned above!. Actu-ally, as with all human beings, Alice and Bob can commnicate only at a classical level, by adjusting the macroscosettings of their measurement apparatuses and observinred and green light flashes associated with the results of msurements. Paraphrasing Bohr~see the end of Sec. I B!, wecould say that ‘‘there is no such concept as quantum inmation; information is inherently classical, but may be tranmitted through quantumq bits;’’ nevertheless, the wholefield is now sometimes called ‘‘quantum informatiotheory.’’ For an early proposal of a practical scheme oquantum computer with cold trapped ions, see Ref. 123.

Decoherence is the big enemy of quantum computatfor a simple reason: It constantly tends to destroy the uscoherent superpositions; this sadly reduces the full quaninformation to its classical, Boolean, component madediagonal matrix elements only. It is now actually perfecclear that a ‘‘crude’’ quantum computer based on the nause of nonredundantq bits will never work, at least withmore than a very small number of them; it has beenmarked that this kind of quantum computer would simplya sort of resurgence of the old analog computers~errors inquantum information form a continuum!, in an especiallyfragile version! But it has also been pointed out that anpropriate use of quantum redundancy may allow one tosign efficient error correcting schemes~Refs. 124 and 125!;decoherence can be corrected by using a system contamoreq bits, and by projecting its state into some subspain which the correct information about the significantq bitsurvives without error~Ref. 126!; the theoretical schemeinvolve collective measurements of severalq bits, which

679F. Laloe

b

ahey

sum

thcatac

ghi

t

nforea

tdea

re

eastn.

histata

ov

iveontaitgndeeer

ola93orn

ury

ole

ersr-butysion

ereaentits:tiesthe

to as.-ityillns:tor,ndt re-al

an-en-enaysthatthe

ional-cal-s-an

ical,llyndeiseri-

rk:o-

yith

es.

er

edty

itsn be

give access to some combined information on all them,none on a singleq bit. It turns out that it is theoreticallypossible to ‘‘purify’’ quantum states by combining seversystems in perturbed entangled states and applying to tlocal operations, in order to extract a smaller number of stems in nonperturbed states~Ref. 127!; one sometimes alsospeaks of ‘‘quantum distillation’’ in this context. Thischeme applies in various situations, including quantcomputation as well as communication or cryptography~Ref.128!. Similarly the notion of ‘‘quantum repeaters’’~Ref.129! has been introduced recently in order to correct foreffect of imperfections and noise in quantum communition. Another very different approach to quantum compution has been proposed, based on a semiclassical conwhere q bits are still used, but communicate only throuclassical macroscopic signals, which are used to determthe type of measurement performed on the nextq bit ~Ref.130!; this kind of computer should be much less sensitivedecoherence.

Generally speaking, whether or not it will be possible oday to beat decoherence in a sufficiently large systempractical quantum computing still remains to be seen. Moover, although the factorization into prime numbers isimportant question~in particular for cryptography!, as wellas the many-body quantum problem, it would be niceapply the principles of quantum computation to a broascope of problems! The question as to whether or not qutum computation will become a practical tool one daymains open to debate~Refs. 131 and 119!, but in any casethis is an exciting new field of research.

VI. VARIOUS INTERPRETATIONS

In Sec. I we have already mentioned some of the oth‘‘unorthodox’’ interpretations of quantum mechanics thhave been proposed, some of them long ago and almoparallel with the ‘‘orthodox’’ Copenhagen interpretatioOur purpose was then to discuss, in a historical context, wthey are now generally taken more seriously by physicthan they were in the middle of the 20th century, but notgive any detail; this article would be incomplete without,least, some introduction to the major alternative interpretions of quantum mechanics that have been proposedthe years, and this is the content of Sec. VI.

It is clearly out of the question to give here an exhaustdiscussion of all possible interpretations. This would evprobably be an impossible task! The reason is that, whilecan certainly distinguish big families among the interpretions, it is also possible to combine them in many ways, wan almost infinite number of nuances. Even the Copenhainterpretation itself is certainly not a monolithic constructioit can be seen from different points of view and can beclined in various forms. An extreme case was already mtioned in Sec. II B: what is sometimes called the ‘‘Wigninterpretation’’ of quantum mechanics, probably becausethe title and conclusion of Ref. 37—but views along similines were already discussed by London and Bauer in 1~Ref. 132!. In this interpretation, the origin of the state vectreduction should be related to consciousness. For instaLondon and Bauer emphasize that state vector reductionstores a pure state from a statistical mixture of the meassubsystem~see Sec. V C 2!, and ‘‘the essential role played bthe consciousness of the observer in this transition frommixture to a pure state;’’ they then explain this special r


ut

lm

s-

e--ept

ne

o

er-

n

orn--

r,tin

ytsot-er

ene-hen;-

n-

fr9

ce,re-ed

a

by the faculty of introspection of conscious observers. Othprefer to invoke ‘‘special properties’’ of the electrical curents which correspond to perception in a human brain,how seriously this explanation is put forward is not alwaentirely clear. In fact, Wigner may have seen the introductof an influence of consciousness just as an extreme case~ex-actly as the Schro¨dinger cat was introduced by Schro¨dinger!,just for illustrating the necessity of a nonlinear step in ordto predict definite results. In any event, the merit of the idis also to show how the logic of the notion of measuremin the Copenhagen interpretation can be pushed to its limIndeed, how is it possible to ascribe such special properto the operation of measurement without considering thathuman mind also has very special properties?

For obvious reasons of space, here we limit ourselvessketchy description of the major families of interpretationWe actually start with what we might call a ‘‘minimal interpretation,’’ a sort of common ground that the vast majorof physicists will consider as a safe starting point. We wthen proceed to discuss various families of interpretatioadditional variables, nonlinear evolution of the state vecconsistent histories, Everett interpretation. All of them teto change the status of the postulate of the wave packeduction; some interpretations incorporate it into the normSchrodinger evolution, others make it a consequence ofother physical process that is considered as more fundamtal, still others use a formalism where the reduction is hiddor even never takes place. But the general purpose alwremains the same: to solve the problems and questionsare associated with the coexistence of two postulates forevolution of the state vector.

A. Common ground; ‘‘correlation interpretation’’

The method of calculation that we discuss in this sectbelongs to standard mechanics; it is actually common tomost all interpretations of quantum mechanics and, as aculation, very few physicists would probably put it into quetion. On the other hand, when the calculation is seen asinterpretation, it may be considered by some as too technand not sufficiently complete conceptually, to be reacalled an interpretation. But others may feel differently, awe will nevertheless call it this way; we will actually use thwords ‘‘correlation interpretation,’’ since all the emphasisput on the correlations between successive results of expments.

The point of view in question starts from a simple remaThe Schro¨dinger equation alone, more precisely its transpsition to the ‘‘Heisenberg point of view,’’ allows a relativelstraightforward calculation of the probability associated wany sequence of measurements, performed at different timTo see how, let us assume that a measurement38 of a physicalquantity associated with operatorM is performed at timet1 ,and callm the possible results; this is followed by anothmeasurement of observableN at time t2 , with possible re-sultsn, etc. Initially, we assume that the system is describby a pure stateuC(t0)&, but below we generalize to a densioperatorr(t0). According to the Schro¨dinger equation, thestate vector evolves between timet0 and time t1 fromuC(t0)& to uC(t1)&; let us then expand this new state intocomponents corresponding to the various results that caobtained at timet1 :

uC~ t1!&5(m

uCm~ t1!&, ~32!

680F. Laloe

hee

sreome

foer

dit

he

r

at.ef

gn

t i

,

nt

avwa

thctv

ioa

si

e,hythis

nerre-ath-of

tt iss isin

onem-

e itical

wea-

conding-

s of, a

aler-lt in

s,rela-e

mentthisveess,ber

in-it.the

-ndeed

ofheini-

tionscalea-

s webepe-w,t ofrms

his

hert asrst

whereuCm(t1)& is obtained by applying touC(t1)& the pro-jector PM(m) on the subspace corresponding to resultm:

uCm~ t1!&5PM~m!uC~ t1!&. ~33!

Now, just after the first measurement, we can ‘‘chop’’ tstate vector into different ‘‘slices,’’ which are each of thterms contained in the sum of~32!. In the future, these termwill never give rise to interference effects, since they corspond to different measurement results; actually, each cponent becomes correlated to an orthogonal state of thevironment ~the pointer of the measurement apparatusinstance! and a full decoherence will ensure that any intference effect is canceled.

Each ‘‘slice’’ uCm(t1)& of uC(t1)& can then be considereas independent from the others, and taken as a new instate of the system under study. From timet1 to time t2 , thestate in question will then evolve under the effect of tSchrodinger equation and become a stateuCm(t2)&. For thesecond measurement, the procedure repeats itself;‘‘slice’’ again this new state according to

uCm~ t2!&5(n

uCm,n~ t2!&, ~34!

where uCm,n(t2)& is obtained by the action of the projectoPN(n) on the subspace corresponding to resultn:

uCm,n~ t2!&5PN~n!uCm~ t2!&. ~35!

The evolution of eachuCm(t2)& will now be considered in-dependently and, if a third measurement is performedlater timet3 , generate one more decomposition, and so onis easy to check39 that the probability of any given sequencof measurementsm, n, p, etc., is nothing but by the square othe norm of the final state vector:

P~m,t1 ;n,t2 ;p,t3 ;..!5u^Cm,n,p,..,q~ tq!uCm,n,p,...,q~ tq!&u2.~36!

Let us now describe the initial state of the system throua density operatorr(t0); it turns out that the same result cabe written in a compact way, according to a formula thasometimes called the Wigner formula~Refs. 38 and 133!.For this purpose, we consider the time-dependent versionthe Heisenberg point of view,40 of all projectors:PM(m;t)corresponds toPM(m), PN(n;t), to PN(n), etc. One canthen show that the probability for obtaining resultm fol-lowed by resultn is given by41

P~m,t1 ;n,t2!

5Tr$PN~n;t2!PM~m;t1!r~ t0!PM~m;t1!PN~n;t2!% ~37!

~generalizing this formula to more than two measuremewith additional projectors, is straightforward!.

Equation~37! can be seen as a consequence of the wpacket reduction postulate of quantum mechanics, sinceobtained it in this way. But it is also possible to take it asstarting point, as a postulate in itself: It then providesprobability of any sequence of measurements, in a perfeunambiguous way, without resorting either to the wapacket reduction or even to the Schro¨dinger equation itself.The latter is actually contained in the Heisenberg evolutof projection operators, but it remains true that a direct cculation of the evolution ofuC& is not really necessary. Afor the wave packet reduction, it is also contained in a waythe trace operation of~37!, but even less explicitly. If one


--

n-r-

ial

we

aIt

h

s

in

s,

ee

elye

nl-

n

just uses formula~37!, no conflict of postulates takes placno discontinuous jump of any mathematical quantity; wnot then give up entirely the other postulates and just usesingle formula for all predictions of results?

This is indeed the best solution for some physicists: If oaccepts the idea that the purpose of physics is only to colate the preparation of a physical system, contained mematically inr(t0), with all possible sequence of resultsmeasurements~by providing their probabilities!, it is truethat nothing more than~37! is needed. Why then worry abouwhich sequence is realized in a particular experiment? Isufficient to assume that the behavior of physical systemfundamentally indeterministic, and that there is no needphysics to do more than just giving rules for the calculatiof probabilities. The ‘‘correlation interpretation’’ is therefora perfectly consistent attitude; on the other hand, it is copletely opposed to the line of the EPR reasoning, sincshows no interest whatsoever in questions related to physreality as something ‘‘in itself.’’ Questions such as: ‘‘hoshould the physical system be described when one first msurement has already been performed, but before the semeasurement is decided’’ should be dismissed as meanless. Needless to say, the notion of the EPR elementreality becomes completely irrelevant, at least to physicslogical situation which automatically solves all potentiproblems related to Bell, GHZ, and Hardy type considations. The same is true of the emergence of a single resua single experiment; in a sense, the Schro¨dinger cat paradoxis eliminated by putting it outside of the scope of physicbecause no paradox can be expressed in terms of cortions. An interesting feature of this point of view is that thboundary between the measured system and the environof the measuring devices is flexible; an advantage offlexibility is that the method is well suited for successiapproximations in the treatment of a measurement procfor instance the tracks left by a particle in a bubble chamas discussed by Bell~Ref. 34!.

In practice, most physicists who favor the correlationterpretation do not feel the need for making it very explicNevertheless, this is not always the case; see for instancearticle by Mermin~Ref. 134!, which starts from the statement: ‘‘Throughout this essay, I shall treat correlations aprobabilities as primitive concepts.’’ In a similar context, salso a recent ‘‘opinion’’ in Physics Today by Fuchs anPeres~Ref. 31! who emphasize ‘‘the internal consistencythe theory without interpretation.’’ On the other hand, tcorrelation interpretation is seen by some physicists as mmalistic because it leaves aside, as irrelevant, a few questhat they find important; an example is the notion of physireality, seen as an entity that should be independent of msurements performed by human beings. Nevertheless, ahave already mentioned, the interpretation can easilysupplemented by others that are more specific. In fact, exrience shows that defenders of the correlation point of viewhen pressed hard in a discussion to describe their poinview with more accuracy, often express themselves in tethat come very close to the Everett interpretation~see Sec.VI E!; in fact, they may sometimes be proponents of tinterpretation without realizing it!

Let us finally mention in passing that formula~37! may bethe starting point for many interesting discussions, whetor not it is considered as basic in the interpretation, or jusa convenient formula. Suppose for instance that the fi

681F. Laloe

ue

o

atreisreatocere

aus

ba

monrathen

dch

eilln

heo

geeh

usgl

tsefsa

exs,oth

rieeson

ar

e-In aosesdfhei-

tro-slsor-

ityly,

thein-ticalater

i-ayyrytracts a

alhedi-usbleg ofac-er

afu-at

i-thettynyofthe

esl-

m,veal

en

po-

-lestin-

measurement is associated with a degenerate eigenvalan operator, in other words thatPM(m;t1) is a projector overa subspace of more than one dimension:

PM~m;t1!5(i 51

n

uw i&^w i u ~38!

~for the sake of simplicity we assume thatt15t0 , so that notime dependence appears in this expression!. Inserting thisexpression into~37! immediately shows the appearanceinterference terms~or crossed terms! iÞ j between the con-tribution of the variousuw i&. Assume, on the other hand, thmore information was actually obtained in the first measument, so that the value ofi was also determined, but that thinformation was lost, or forgotten; the experimenter ignowhich of two or morei results was obtained. Then, whshould be calculated is the sum of the probabilities assated with each possible result, that is a single sum ovifrom which all crossed termsiÞ j have disappeared. In thfirst case, interference terms arise because one has toprobability amplitudes; in the second, they do not becaone has to add the probabilities themselves~exclusiveevents!. The contrast between these two situations mayunderstood in terms of decoherence: In the first case,states of the system correlate to the same state of thesuring apparatus, which here plays the role of the envirment; they do not in the second case, so that by partial tall interference effects vanish. This remark is useful indiscussion of the close relation between the so-called ‘‘Zparadox in quantum mechanics’’~Ref. 135! and decoher-ence; it is also basic in the definition of consistency contions in the point of view of decoherent histories, to whiwe will come back later~Sec. VI D!.

B. Additional variables

We now leave the range of considerations that are morless common to all interpretations; from now on, we wintroduce in the discussion some elements that clearly dobelong to the orthodox interpretation. We begin with ttheories with additional variables, as the de Broglie theorythe pilot wave~Ref. 136!; the work of Bohm is also knownas a major reference on the subject~Refs. 9 and 137!; seealso the almost contemporary work of Wiener and Sie~Ref. 10!. More generally, with or without explicit referencto additional variables, one can find a number of authors wsupport the idea that the quantum state vector should beonly for the description of statistical ensembles, not of sinevents, see for instance Refs. 138 and 46.

We have already emphasized that the EPR theorem ican be seen as an argument in favor of the existencadditional variables~we will come back later to the impact othe Bell and BKS theorems!. Theories with such variablecan be built mathematically in order to reproduce exactlypredictions of orthodox quantum mechanics; if they giveactly the same probabilities for all possible measurementis clear that there is no hope to disprove experimentallythodox quantum mechanics in favor of these theories, oropposite. In this sense they are not completely new theobut rather variations on a known theory. They neverthelhave a real conceptual interest: They can restore notrealism, but also determinism~this is a possibility but not anecessity—one can also build theories with additional vables that remain fundamentally nondeterministic!.


of

f

-

s

i-

dde

ell

ea--

ceeo

i-

or

ot

f

l

oede

elfof

ll-it

r-es,sly

i-

1. General framework

None of the usual ingredients of orthodox quantum mchanics disappears in theories with additional variables.sense, they are even reinforced, since the wave function lits subtle status~see Sec. I B!, and becomes an ordinary fielwith two components~the real part and the imaginary part othe wave function—for simplicity, we assume here that tparticle is spinless!; these components are for instance simlar to the electric and magnetic components of the elecmagnetic field.42 The Schro¨dinger equation itself remainstrictly unchanged. But a completely new ingredient is aintroduced: In addition to its wave function field, each paticle gets an additional variablel, which evolves in timeaccording to a new equation. The evolution ofl is actuallycoupled to the real field, through a sort of ‘‘quantum velocterm’’43 that depends on the wave function; but, conversethere is no retroaction of the additional variables ontowave function. From the beginning, the theory thereforetroduces a marked asymmetry between the two mathemaobjects that are used to describe a particle; we will see lthat they also have very different physical properties.

For anyone who is not familiar with the concept, addtional variables may look somewhat mysterious; this mexplain why they are often called ‘‘hidden,’’ but this is onla consequence of our much better familiarity with ordinaquantum mechanics! In fact, these variables are less absthan the wave functions, which in these theories becomesort of auxiliary field, even if perfectly real. The additionvariables are directly ‘‘seen’’ in a measurement, while tstate vector remains invisible; it actually plays a rather inrect role, through its effect on the additional variables. Lettake the example of a particle creating a track in a bubchamber: On the photograph we directly see the recordinthe successive values of an additional variable, which istually nothing but...the position of the particle! Who has evtaken a photograph of a wave function?

For a single particle, the additional variablel may there-fore also be denoted asR since it describes its position; formany particle system,l is nothing but a symbol for the set opositionsR1 ,R2 , etc., of all the particles. The theory postlates an initial random distribution of these variables thdepends on the initial wave functionC(r1 ,r2 ,...) andrepro-duces exactly the initial distribution of probability for postion measurements; using hydrodynamic versions ofSchrodinger equation~Ref. 139!, one can easily show thathe evolution under the effect of the ‘‘quantum velociterm’’ ensures that this property continues to be true for atime. This provides a close contact with all the predictionsquantum mechanics; it ensures, for instance, that undereffect of the quantum velocity term the position of particlwill not move independently of the wave function, but aways remains inside it.

At this point, it becomes natural to restore determinisand to assume that the results of measurements merely rethe initial pre-existing value ofl, chosen among all possiblvalues in the initial probability distribution. This assumptiosolves many difficulties, all those related to the Schro¨dingercat paradox for instance: Depending on the exact initialsition of a many-dimension variablel, which belongs to anenormous configuration space~including the variables associated with the radioactive nucleus as well as all variabassociated with the cat!, the cat remains alive or dies. Burestoring determinism is not compulsory, and nondeterm

682F. Laloe

en

yrn

o

thxi

ng-

es

at

orth

aleracar-

onr

ce

etetulttuthe

ttieunc

ioe

efeta

o

,a

al

e

pthT

gheon

ofar-thesi-le

ong

vepa-

ix-the

dsyics!

aten

-

us

eceheaneticsed-er-hery

omenceerianatr oftheri-esttedx

thear-field

.’’fbetterathe

le:rstv-he

istic versions of additional variables can easily be designIn any case, the theory will be equivalent to ordinary quatum mechanics; for instance, decoherence will act exactlthe same way, and make it impossible in practice to obseinterferences with macroscopic objects in very differestates.

To summarize, we have in this context a descriptionphysical reality at two different levels:

~i! one corresponding to the elements associated withstate vector, which can be influenced directly in eperiments, since its evolution depends on a Hamtonian that can be controlled, for instance by applyifields; this level alone is not sufficient to give a complete description of a physical system,

~ii ! another corresponding to the additional variablwhich cannot be manipulated directly~see AppendixE!, but obey evolution equations containing the stvector.

The two levels together are necessary and sufficient fcomplete description of reality. There is no retroaction ofadditional variables onto the state vector, which createsunusual situation in physics~usually, when two physicaquantities are coupled, they mutually influence each oth!.Amusingly, we are now contemplating another sort of duity, which distinguishes between direct action on physisystems~or preparation! and results of observations peformed on them~results of measurements!.

A similar line of thought has been developed by Nels~Ref. 140!, who introduces stochastic motions of point paticles in such a way that their statistical behavior reproduexactly the predictions of the Schro¨dinger equation. The dif-ference is that the evolution of the wave function is not givby a postulate, but is actually derived from other postulathat are considered more fundamental. This leads to a naderivation of the Schro¨dinger equation; the formalism is buito lead exactly to the same predictions as orthodox quanmechanics, so that its interest is mostly conceptual. Fordiscussion of statistical mixtures in this context, see R141.

2. Bohmian trajectories

As soon as particles regain a position, they also getrajectory, so that it becomes natural to study their properin various situations; actually one then gets a variety ofexpected results. Even for a single particle in free spabecause of the effects of its wave function on the evolutof its position, it turns out that the trajectories are not necsarily simple straight lines~Ref. 142!; in interference experi-ments, particles may actually follow curved trajectories evin regions of space where they are free, an unusual efindeed.44 But this feature is in fact indispensable for the stistics of the positions to reproduce the usual predictionsquantum mechanics~interference fringes!, Ref. 143. Bellstudied these questions~Ref. 34! and showed, for instancethat the observation of successive positions of a particlelows one to reconstruct a trajectory that remains physicacceptable.

For systems of two particles or more, the situation bcomes even more interesting. Since the Schro¨dinger equationremains unchanged, the wave functions continue to progate in the configuration space, while on the other handpositions propagate in ordinary three-dimensional space.


d.-invet

f

e-l-

,

e

aen

l-l

-s

nsral

me

f.

as-e,ns-

nct-f

l-ly

-

a-e

he

effects of nonlocality become especially apparent throuthe ‘‘quantum velocity term,’’ since the velocity has to bevaluated at a point of configuration space that dependsthe positions of both particles; the result for the velocityparticle 1 may then depend explicitly on the position of pticle 2. Consider for instance an EPRB experiment oftype described in Sec. IV A 2 and the evolution of the potions of the two particles when they are far apart. If partic1 is sent through a Stern–Gerlach analyzer oriented aldirectiona, the evolution of its Bohmian position will obvi-ously be affected in a way that depends ona ~we remarkedabove that the positions have to follow the quantum wafunctions; in this case, it has the choice between two serating wave packets!. But this will also change the position(R1 ,R2) of the point representing the system in the sdimensional configuration space, and therefore changequantum velocity term for particle 2, in a way that depenexplicitly on a. No wonder if such a theory has no difficultin reproducing the nonlocal features of quantum mechanThe advantage of introducing additional variables is, insense, to emphasize the effects of nonlocality, which ofremain relatively hidden in the orthodox formalism~onemore reason not to call these variables ‘‘hidden!’’!. Bell forinstance wrote ‘‘it is a merit of the Broglie–Bohm interpretation to bring this~non-locality! out so explicitly that it cannot be ignored’’—in fact, historically, he came to his famoinequalities precisely through this channel.

An interesting illustration of this fact can be found in thstudy of Bohmian trajectories in a two-particle interferenexperiment, or in a similar case studied in Ref. 144. Tauthors of this reference study a situation which involvesinterference experiment supplemented by electromagncavities, which can store the energy of photons and be uas a ‘‘Welcher Weg’’ device~a device that tells the experimenter which hole the particle went through in an interfence experiment!. A second particle probes the state of tfield inside the cavity, and when leaving it takes a trajectothat depends on this field. These authors show that, in sevents, a particle can leave a photon in a cavity and influea second particle, while the trajectory of the latter nevcrosses the cavity; from this they conclude that the Bohmtrajectories are ‘‘surrealistic.’’ Of course, considering thtrajectories are surrealistic or not is somewhat a mattetaste. What is clear, however, is that a firm believer inBohmian interpretation will not consider this thought expement as a valid argument against this interpretation—at bhe/she will see it as a valid attack against some truncaform of Bohmian theory. One should not mix up orthodoand Bohmian theories, but always keep in mind that, inlatter theory, the wave function has a totally different chacter: It becomes a real classical field, as real as a laserfor instance. As expressed by Bell~Ref. 145!: ‘‘No one canunderstand this theory until he is willing to think ofC as areal objective field rather than just a probability amplitudeTherefore, a ‘‘particle’’ always involves a combination oboth a position and the associated field, which cannotdissociated; there is no reason whatsoever why the lacould not also influence its surrounding. It would thus bemistake to assume that influences should take place invicinity of the trajectory only.

In this context, the way out of the paradox is then simpjust to say that the real field associated with the fiparticle45 interacted with the electromagnetic field in the caity, leaving a photon in it; later this photon acted on t

683F. Laloe

t ive

twn

mfelexocc

tinnceoseheba

ootsesh

e-ag,heoiv

eet

tiohe

ei-

sthinteinethe

rnteif

datnyth

ic

con-.

di-ro-rsub

orsly

e-dof

cs.eit ise’’x-en

nofodi-

tumthesesgo

opicitialicalperi-this

ions.s a

as aeees

r-nents

s lo-

ts infor

ddi-ofll astionscalti-

thened

y--in

trajectory of the second particle. In other words, the effeca crossed field-trajectory effect, and in these terms it is eperfectly local! One could even add that, even if for somreason one decided to just consider the trajectories of theparticles, the fact that they can influence each other, evethey never come close to each other creates no probleitself; it is just an illustration of the explicit character ononlocality in the Bohm theory—see the quotation by Babove, as well as the discussion of this thought experimby Griffiths in Ref. 146. So, we simply have one more eample of the fact that quantum phenomena are indeed lin configuration space, but not necessarily in ordinary spa

This thought experiment nevertheless raises interesquestions, such as: if in this example a particle can influeevents outside of its own Bohmian trajectory, what is ththe physical meaning of this trajectory in general? Suppthat, in a cloud chamber for instance, a particle could leavtrack that does not coincide at all with the trajectory of tBohmian position; in what sense then could this variablecalled ‘‘position’’? For the moment, that this strange sitution can indeed occur has not been shown~the exampletreated in Ref. 144 is very special and presumably not a gmodel for a cloud chamber!, but the question clearly requesmore precise investigation. Another difficulty of theoriwith additional variables is also illustrated by this thougexperiment: the necessity for including fields~in this case thephotons in the cavities!. Quantum mechanics is used to dscribe a large variety of fields, from the usual electromnetic field~quantum electrodynamics! to quarks, for instanceand this is truly essential for a physical description of tworld; at least for the moment, the complete descriptionall these fields has not been developed within theories wadditional variables, although attempts in this direction habeen made.

C. Modified „nonlinear… Schrodinger dynamics

Another way to resolve the coexistence problem betwthe two postulates of quantum mechanics is to changeSchrodinger equation itself: One assumes that the equaof evolution of the wave function contains, in addition to tusual Hamiltonian terms, nonlinear~and possibly stochastic!terms, which will also affect the state vector~Refs. 4, 11, 13,147, and 148!. These terms may be designed so that theffects remain extremely small in all situations involving mcroscopic objects only~atoms, molecules, etc.!; this will im-mediately ensure that all the enormous amount of succespredictions of quantum mechanics is capitalized. Onother hand, for macroscopic superpositions involving, forstance, pointers of measurement apparatuses, the newmay mimic the effects of wave packet reduction, by selectone branch of the superposition and canceling all the othClearly, one should avoid both extremes: either perturbSchrodinger equation too much, and make interferencefects disappear while they are still needed~for instance, pos-sible recombination of the two beams at the exit of a SteGerlach magnet!; or too little, and not ensure the compledisappearance of Schro¨dinger cats! This result is obtainedthe perturbation term becomes efficient when~but not be-fore! any microscopic system becomes strongly correlatea macroscopic environment, which ensures that significdecoherence has already taken place; we then know tharecovery of interference effects is impossible in practice away. If carefully designed, the process then reproduceseffect of the postulate of the wave function collapse, wh


sn

eoifin

lnt-ale.ge

nea

e-

d

t

-

fthe

nhen

ir

fule-rmsgrs.ef-

–

tontthe-e

h

no longer appears as an independent postulate, but as asequence of the ‘‘normal’’ evolution of the wave function

1. Various forms of the theory

There are actually various versions of theories with mofied Schro¨dinger dynamics. Some versions request the intduction of additional variables into the theory, while othedo not. The approach proposed in 1966 by Bohm and B~Ref. 4! belongs to the first category, since these authincorporate in their theory additional variables previousconsidered by Wiener and Siegel~Ref. 10!; these variablesare actually contained in a ‘‘dual vector,’’ similar to thusual state vectoruC&, but obeying an entirely different equation of motion—in fact, both vectors evolve with coupleequations. What is then obtained is a sort of combinationtheories with additional variables and modified dynamiFor some ‘‘normal’’ distribution of the new variables, thprediction of usual quantum mechanics is recovered; butalso possible to assume the existence of ‘‘dispersion fredistributions that lead to nonorthodox predictions. An eample of models that are free of additional variables is givby the work of Pearle~Ref. 11!, published ten years later, iwhich nothing is added to the usual conceptual framestandard quantum mechanics. The theory is based on a mfied dynamics for the modulus and phases of the quanamplitudes, which get appropriate equations of evolution;result is that, depending on the initial values of the phabefore a measurement, all probability amplitudes but oneto zero during a measurement. Because, when a microscsystem is sent toward a macroscopic apparatus, the inphases are impossible to control with perfect mathemataccuracy, an apparent randomness in the results of exments is predicted; the equations are designed so thatrandomness exactly matches the usual quantum predictIn both theories, the reduction of the state vector becomedynamical process which, as any dynamical process, hfinite time duration; for a discussion of this question, sRef. 149, which remarks that the theory of Ref. 4 introducan infinite time for complete reduction.

Another line of thought was developed from consideations that were initially not directly related to wave functiocollapse, but to continuous observations and measuremin quantum mechanics~Refs. 150 and 151!. This was thestarting point for the work of Ghirardiet al. ~Ref. 13!, whointroduce a random and sudden process of ‘‘spontaneoucalization’’ with an arbitrary frequency~coupling constant!,which resembles the effect of approximate measuremenquantum mechanics. The constant is adjusted so that,macroscopic systems~and for them only!, the occurrence ofsuperposition of far-away states is destroyed by the ational process; the compatibility between the dynamicsmicroscopic and macroscopic systems is ensured, as wethe disappearance of macroscopic coherent superposi~transformation of coherent superpositions into statistimixtures!. This approach solves problems that were idenfied in previous work~Ref. 11!, for instance the ‘‘preferredbasis problem,’’ since the basis is that of localized states;relation to the quantum theory of measurement is examiin detail in Ref. 152. In this model, for individual systems46

the localization processes are sudden~they are sometimescalled ‘‘hitting processes’’!, which makes them completeldifferent from the usual Schro¨dinger dynamics. Nevertheless, later work~Ref. 153! showed that it is possible to design theories involving only continuous evolution that reta

684F. Laloe

oa

on

a-

n-

s

rees

nio

naceaRhait

eeespealbe

-fo

heicg

caheabhac

ubveisnoe

prn

blee

b-hebeia-x-thf o-

not

larpar-the

ment

onal

atenta,eenctu-silyallctsd.at

ofzedece;ear

isentba-

lrn–

ctlyasthein

el isere,n

eslso

-it isvehe-x-

Ifanded.ome

vethereibil-the

the attractive features of the model. For instance, the disctinuous Markov processes in Hilbert space reduce, in anpropriate limit, to a continuous spontaneous localizatiwhich may result in a new version of nonlinear Schro¨dingerdynamics~Ref. 154! called continuous spontaneous localiztion ~CSL!; another achievement of Ref. 154 is a full compatibility with the usual notion of identical particles in quatum mechanics. See also Ref. 147 for an earlier versionmodified Schro¨dinger dynamics with very similar equationof evolution.

A similar line was followed by Diosi~Ref. 148!, who alsostarted initially from the treatment of continuous measuments~Ref. 155! by the introduction of stochastic process~‘‘quantum Wiener processes,’’ Ref. 10! that are added to theusual deterministic Schro¨dinger dynamics. This author theintroduced a treatment of the collapse of the wave functfrom a universal law of density localization~Ref. 156!, witha strength that is proportional to the gravitational constaresulting in a parameter free unification of micro- and mrodynamics. Nevertheless, this approach was found to crsevere problems at short distances by the authors of157, who then proposed a modification of the theory tsolves them, but at the price of reintroducing a constant wdimension~a length!.

Generally speaking, beyond their fundamental purpos~aunification of all kinds of physical evolution, including wavfunction reduction!, two general features of these theorishould be emphasized. The first is that new constants apwhich may in a sense look like ad hoc constants, but actuhave an important conceptual role: They define the limittween the microscopic and macroscopic world~or betweenreversible and irreversible evolution!; the corresponding border is no longer ill-defined, as opposed to the situation,instance, in the Copenhagen interpretation. The second~re-lated! feature is that these theories are more predictive. Tare actually the only ones which propose a real physmechanism for the emergence of a single result in a sinexperiment, which is of course attractive from a physipoint of view. At the same time, and precisely because tare more predictive, these theories become more vulnerto falsification, and one has to carefully design the mecnism in a way that satisfies many constraints. For instanwe have already mentioned that, in the initial Bohm–Btheory, a complete collapse of the wave function is neobtained in any finite time. The same feature actually exin CSL: There is always what is called a ‘‘tail’’ and, evewhen most of the wave function goes to the component cresponding to one single outcome of an experiment, thalways remains a tiny component on the others~extremelysmall and continuously going down in size!. The existence ofthis component is not considered as problematic by theponents of the CSL theory, as illustrated by the contributioof Pearle and Ghirardi in Ref. 158. In the context of possiconflicts with experiments, see also the discussion of R157 concerning incompatibilities of another form of ththeory with the well-known properties of microscopic ojects, as well as Ref. 159 for a critical discussion of anotversion of nonlinear dynamics. A similar case is providedthe generalization of quantum mechanics proposed by Wberg ~Ref. 160!, which this author introduced as an illustrtion of a nonlinearity that is incompatible with available eperimental data; see also Ref. 161 for an application ofsame theory to quantum optics and Ref. 162 for a proothe incompatibility of this theory relativity, due to the pre


n-p-,

-

of

-

n

t,-te

ef.th

ar,ly-

r

yallelyle-

e,

rts

r-re

o-sef.

ryn-

ef

diction of superluminal communication~the proof is specificto the Weinberg form of the nonlinear theory and doesapply to the other forms mentioned above!.

2. Physical predictions

Whatever specific form of the theory is preferred, simiphysical descriptions are obtained. For instance, when aticle crosses a bubble chamber, the new terms createappearance~at a macroscopic level! of a particle trajectory;they also select one of the wave packets at the measureoutput of a Stern–Gerlach analyzer~and eliminate the other!,but not before these packets become correlated to orthogstates of the environment~e.g., detectors!. Of course, anyprocess of localization of the wave function tends to operin the space of positions rather than in the space of momewhich reduces to some extent the usual symmetry betwpositions and momenta in quantum mechanics. This is aally not a problem, but a convenient feature: One can eaconvince oneself that, in practice, what is measured inexperiments is basically the positions of particles or obje~pointers, etc.!, while momenta are only indirectly measureGenerally speaking, it is a different spatial localization thproduces wave packet collapse.

How is an EPRB experiment described in this pointview? In the case of Bohmian trajectories, we emphasithe role of the ‘‘quantum velocity term,’’ which has a valudefined in configuration space and not in ordinary spahere, what is essential is the role of the added nonlinlocalization term in the Schro¨dinger equation, which alsoacts in the six-dimensional configuration space. This termdesigned so that, when correlation with the environmtakes place, one of the components in the correspondingsis ~‘‘basis of decoherence’’! is selected. Nothing speciathen occurs as long as particle 1 propagates within a SteGerlach analyzer, since it is microscopic and can perfewell go through superpositions of far-away states; butsoon as it hits a detector at the output of the magnet,system develops correlations with the particles containedthe detector, the amplifier, etc., so that a macroscopic levreached and the localization term becomes effective. Hwe see that it is thea dependence of the spatial localizatio~in other words, the basis of decoherence! that introduces anoverall effect on the two-particle state vector; it providparticle 2 with, not only a privileged spin-state basis, but aa reduction of its spin state to one single component~whenparticle 1 hits the detector!. Since this point of view emphasizes the role of the detectors and not of the analyzers,clearly closer to the usual interpretation, in terms of wapacket reduction, than the Bohmian interpretation. Nevertless, it also puts into light the role of nonlocality in an eplicit way, as this interpretation does.

What about the Schro¨dinger cat and similar paradoxes?the added nonlinear term has all the required propertiesmimics the wave packet reduction, they are easily solvFor instance, a broken poison bottle must have at least sparts that have a different spatial localization~in configura-tion space! than an unbroken bottle; otherwise it would haall the same physical properties. It is then clear thatmodified dynamics will resolve the components long befoit reaches the cat, so that the emergence of a single possity is ensured. For a recent discussion of the effects ofmodified dynamics on ‘‘all or nothing coherent states’’~Sec.

685F. Laloe

o

itian-

e

ntctioereagpic

cenibv

: Iio

nteomthrix

apavat

ebyifier

e-st

lie

onc

d

wrethtethed

thero-

e

aces

eswA

torriesbers;

num-acefew

noe-

tois

eve,jec-

pro-ymf

V C 1! in the context of quantum optics, and of the effectsperception in terms of the ‘‘relative state of the brain’’~Sec.VI E!, see Ref. 163.

The program can be seen as a sort of revival of the inhopes of Schro¨dinger, where all relevant physics was cotained in the wave function and its progressive evolution~seethe end of Sec. I A 2!; this is especially true, of course, of thversions of nonlinear dynamics that are continuous~even iffluctuating extra quantities may be introduced!, and not somuch of versions including ‘‘hits’’ that are too reminisceof the wave packet reduction. Here, the state vector diredescribes the physical reality, in contrast with our discussof Sec. I B; we have a new sort of wave mechanics, whthe notion of point particles is given up in favor of tiny wavpackets. The theory is different from theories with additionvariables, because the notion of precise position in confiration space never appears. As we have seen, another imtant difference is that these theories with modified dynamare really new theories: They may, in some circumstanlead to predictions that differ from those of orthodox quatum mechanics, so that experimental tests might be possWe should emphasize that, in this point of view, the wafunction can still not be considered as an ordinary fieldcontinues to propagate in a high dimension configuratspace instead of the usual three dimension space.

A mild version of these theories is found in a variawhere the Schro¨dinger equation remains exactly the sambut where stochastic terms are introduced as a purely cputational tool, and without any fundamental purpose, forcalculation of the evolution of a partial trace density matdescribing a subsystem~Refs. 164–166!; in other words, amaster equation for a density operator is replaced by anerage over several state vectors submitted to a randomturbation, which may in some circumstances turn out to scomputing time very efficiently. Another line of thought thcan be related to some extent to modified Schro¨dinger dy-namics is the ‘‘transactional interpretation’’ of quantum mchanics~Ref. 167!, where a quantum event is describedthe exchange of advanced and retarded waves; as in modnonlinear Schro¨dinger dynamics, these waves are then intpreted as real, and nonlocality is made explicit.

D. History interpretation

The interpretation of ‘‘consistent histories’’ is also somtimes called ‘‘decoherent history interpretation,’’ or ju‘‘history interpretation’’ as we prefer to call it here~becausethe notion of consistency is essential at the level of famiof histories, rather than at the level of individual histories!. Itproposes a logical framework that allows the discussionthe evolution of a closed quantum system, without refereto measurements. The general idea was introduced andveloped by Griffiths~Ref. 15! but it has also been used, ansometimes adapted, by other authors~Refs. 168–170!. Sincethis interpretation is the most recent among those thatdiscuss in this article, we will examine it in somewhat modetail than the others. We will nevertheless remain withinlimits of a nonspecialized introduction; the reader interesin more precise information on the subject should go toreferences that are provided—see also a recent articlPhysics Today~Ref. 171! and the references containetherein.


n

l

lyne

lu-or-ss,-le.etn

,-

e

v-er-e

-

ed-

s

fe

de-

e

edein

1. Histories, families of histories

Consider any orthogonal projectorP on a subspaceF ofthe space of states of a system; it has two eigenvalues,11corresponding to all the states belonging toF, and 0 corre-sponding to all states that are orthogonal toF ~they belong tothe supplementary subspace, which is associated withprojectorQ512P!. One can associate a measurement pcess withP: If the result of the measurement is11, the stateof the system belongs toF; if it is zero, it is orthogonal toF.Assume now that this measurement is made at timet1 on asystem that is initially~at time t0! described by a densityoperatorr(t0); the probability for finding the state of thsystem inF at time t1 is then given by formula~37!, whichin this case simplifies to

P~F,t1!5Tr$P~ t1!r~ t0!P~ t1!%. ~39!

This result can obviously be generalized to several subspF1 ,F2 ,F3 , etc., and several measurement timest1 ,t2 ,t3 ,etc. ~we assumet1,t2,t3,¯!. The probability that thestate of the system belongs toF1 at time t1 , then toF2 attime t2 , then to F3 at time t3 , etc., is, according to theWigner formula,

P~F1 ,t1 ;F2 ,t2 ;F3 ,t3¯ !

5Tr$¯ P3~ t3!P2~ t2!P1~ t1!r~ t0!

3 P1~ t1!P2~ t2!P3~ t3!¯%, ~40!

where, as above, thePi(t i) are the projectors over subspacF1 ,F2 ,F3 in the Heisenberg point of view. We can noassociate a ‘‘history’’ of the system with this equation:historyH is defined by a series of arbitrary timest i , each ofthem associated with an orthogonal projectorPi over anysubspace; its probability is given by~40! which, for simplic-ity, we will write as P~H!. In other words, a history is theselection of a particular path, or branch, for the state vecin a Von Neumann chain, defined mathematically by a seof projectors. Needless to say, there is an enormous numof different histories, which can have all sorts of propertiesome of them are accurate because they contain a largeber of times associated with projectors over small subspF’s; others remain very vague because they contain atimes only with projectors over large subspaceF’s ~one caneven decide thatF is the entire states of spaces, so thatinformation at all is contained in the history at the corrsponding time!.

There are in fact so many histories that it is usefulgroup them into families, or sets, of histories. A familydefined again by an arbitrary series of timest1 ,t2 ,t3 ,..., butnow we associate with each of these timest i an ensemble oforthogonal projectorsPi , j that, when summed, restore thwhole initial space of states. For each time we then hainstead of one single projector, a series of orthogonal protors that provide a decomposition of the unity operator:

(j

Pi , j51. ~41!

This gives the system a choice, so to say, among manyjectors for each timet i , and therefore a choice among manhistories of the same family. It is actually easy to see fro~41! and~40! that the sum of probabilities of all histories oa given family is equal to one:

686F. Laloe

y

y,to

y,il

he

of

oretouoot

th

veth

oc

r-

-?hetioieroan

ctumo

ana,wal

isbthe

r-

tic

om-ndi-

to

t

ce-f

on

r

ex-parttedwreta-

con-anyoiceare

to

e,e toes-

epro-ose

t theov-s atr-

isp-venoci-

d

to aaionhe

tonfer-a-n-

(histories of a family

P~H!51, ~42!

which we interpret as the fact that the system will alwafollow one, and only one, of them.

A family can actually also be built from a single historthe simplest way to incorporate the history into a family isassociate, at each timet i ( i 51,2,...,N), in addition to theprojector Pi , the supplementary projectorQi512Pi ; thefamily then contains 2N individual histories. Needless to sathere are many other ways to complement to single famwith ‘‘more accurate’’ histories than those containing tQ’s; this can be done by decomposing eachQ into manyindividual projectors, the only limit being the dimensionthe total space of states.

2. Consistent families

All this looks very simple, but in general it is actually tosimple to ensure a satisfactory logical consistency in thesonings. Having chosen a given family, it is very naturalalso enclose in the family all those histories that can be bby replacing any pair of projectors, or actually any groupprojectors, by their sum; this is because the sum of twothogonal projectors is again a projector~onto a subspace thais the direct sum of the initial subspaces!. The differenceintroduced by this operation is that, now, at each time,events are no longer necessarily exclusive;47 the historiesincorporate a hierarchy in their descriptive accuracy, eincluding cases where the projector at a given time is justprojector over the whole space of states~no information at allon the system at this time!.

Consider the simplest case where two projectors only,curring at timet i , have been grouped into one single projetor to build a new history. The two ‘‘parent’’ histories corespond to two exclusive possibilities~they containorthogonal projectors!, so that their probabilities add independently in the sum~42!. What about the daughter historyIt is exclusive of neither of its parents and, in terms of tphysical properties of the system, it contains less informaat time t i : The system may have either of the propertassociated with the parents. But a general theorem in pability theory states that the probability associated toevent than can be realized by either of two exclusive eveis the sum of the individual probabilities; one then expethat the probability of the daughter history should be the sof the parent probabilities. On the other hand, inspection~40! shows that this is not necessarily the case; sinceprojector,P2(t2) for instance, appears twice in the formulreplacing it by a sum of projectors introduces four terms: tterms that give the sum of probabilities, as expected, buttwo crossed terms~or ‘‘interference terms’’! between theparent histories, so that the probability of the daughter htory is in general different from the sums of the parent proabilities. These crossed terms are actually very similar toright-hand side of~40!, but the trace always contains at somtime t i one projectorPi , j (t i) on the left of r(t0) and an

orthogonal projectorPi ,k(t i) on the right. This difficulty wasto be expected: We know that quantum mechanics is lineathe level of probability amplitudes, not probabilities themselves; interferences occur because the state vector att i , in the daughter story, may belong to one of the subspa


s

y

a-

iltfr-

e

ne

c--

nsb-ntss

fy

oso

--e

at

mees

associated with the parents, but may also be any linear cbination of such states. As a consequence, a linearity cotion for probabilities is not trivial.

One way to restore the additivity of probabilities isimpose the condition:

Tr$¯ P3,j 3~ t3!P2,j 2

~ t2!P1,j 1~ t1!r~ t0!P1,j

18~ t1!

3 P2,j28~ t2!P3,j

38~ t3!¯%}d j 1 , j

183d j 2 , j

2i 3d j 3 , j

383... .

~43!

Because of the presence of the product ofd’s on the right-hand side, the left-hand side of~43! vanishes as soon as aleast one pair of the indices (j 1 , j 18),( j 2 , j 28),( j 3 , j 38), etc.,contains different values; if they are all equal, the tramerely gives the probabilityP(H) associated with the particular history of the family. What is important for the rest othe discussion is the notion of consistent family: If conditi~43! is fulfilled for all projectors of a given family of histo-ries, we will say that this family is logically consistent, oconsistent for short. Condition~43! is basic in the historyinterpretation of quantum mechanics; it is sometimespressed in a weaker form, as the cancellation of the realonly of the left-hand side; this, as well as other points relato this condition, is briefly discussed in Appendix F. We nodiscuss how consistent families can be used as an interption of quantum mechanics.

3. Quantum evolution of an isolated system

Let us consider an isolated system and suppose that asistent family of histories has been chosen to describe it;consistent family may be selected but, as soon as the chis made, it cannot be modified and all the other familiesexcluded~we discuss later what happens if one attemptsdescribe the same system with more than one family!. Thisunique choice provides us with a well-defined logical framand with a series of possible histories that are accessiblthe system and give information at all intermediate timt1 ,t2 ,... . Which history will actually occur in a given realization of the physical system is not known in advance: Wpostulate the existence of some fundamentally randomcess of Nature that selects one single history among all thof the family. The corresponding probabilityP(H) is givenby the right-hand side of~40!; since this formula belongs tostandard quantum mechanics, this postulate ensures thastandard predictions of the theory are automatically recered. For each realization, the system will then posseseach timet i all physical properties associated with the paticular projectorsPi , j that occur in the selected history. Thprovides a description of the evolution of its physical proerties that can be significantly more accurate than that giby its state vector; in fact, the smaller the subspaces assated with the projectorsPi , j ’s, the more accuracy is gaine~obviously, no information is gained if allPi , j ’s are projec-tors over the whole space of states, but this correspondstrivial case of little interest!. For instance, if the system isparticle and if the projector is a projector over some regof space, we will say that the particle is in this region at tcorresponding time, even if the whole Schro¨dinger wavefunction extends over a much larger region. Or, if a phostrikes a beam splitter, or enters a Mach–Zehnder interometer, some histories of the system may include informtion on which trajectory is chosen by the photon, while sta

687F. Laloe

serath

eaecib

isatun

s-

chthpe

er

ueetit

c

ncy.ieret

lyic

utaoysatherc

cesy

ioie-bases

xtive

use,tent

ce

-his

ntclu-thetentutt isheys-milysid-ly

nten173son

ef.o-

ajectto

theon-

eret ap-

the

edher

c,a

onta-

er-nlyore

thephi-tons;jec-

ruc-

dard quantum mechanics considers that the particle takeof them at the same time. Since histories contain sevdifferent times, one may even attempt to reconstruct anproximate trajectory for the particle, even in cases whereis completely out of the question in standard quantum mchanics~for instance, for a wave function that is a sphericwave!; but of course one must always check that the projtors that are introduced for this purpose remain compatwith the consistency of a family.

In general, the physical information contained in the htories is not necessarily about position only: A projector calso project over a range of eigenstates of the momenoperator, include mixed information on position and mometum ~subject, of course, to Heisenberg relations, as alwayquantum mechanics!, information on spin, etc. There is actually a huge flexibility on the choice of projectors; for eachoice, the physical properties that may be ascribed tosystem are all those that are shared by all states of thejection subspace, but not by any orthogonal state. A frequchoice is to assume that, at a particular timet i , all Pi , j ’s arethe projectors over the eigenstates of some Hermitian optor H: the first operatorPi ,, j 51 is the projector over all theeigenstates ofH corresponding to the eigenvalueh1 , thesecondPi ,, j 52 the corresponding projector for the eigenvalh2 , etc. In this case, all histories of the family will includexact information about the value of the physical quanassociated at timet i to H ~for instance the energy ifH is theHamiltonian!. Let us nevertheless caution the reader onmore that we are not free to choose any operatorHi at anytime t i : In general, there is no reason why the consisteconditions should be satisfied by a family built in this wa

Using histories, we obtain a description of the propertof the system in itself, without any reference to measuments, conscious observers, etc. This does not meanmeasurements are excluded; they can be treated mereparticular cases, by incorporating the corresponding physdevices in the system under study. Moreover, one attribproperties to the system at different times; this is in contrwith the orthodox interpretation; where a measurement dnot necessarily reveal any pre-existing property of the phcal system, and projects it into a new state that may be totindependent of the initial state. It is easy to show thatwhole formalism of consistent families is invariant undtime reversal, in other words that it makes no differenbetween the past and the future@instead of the initial densityoperatorr(t0), one may use the final density operatorr(tN)and still use the same quantum formalism of Ref. 172#—formore details, and even an intrinsic definition of consistenthat involves no density operator at all, see Sec. III of R173. In addition, one can develop a relation between content families and semiclassical descriptions of a physical stem; see Ref. 169 for a discussion of how classical equatcan be recovered for a quantum system provided sufficcoarse graining is included~in order to ensure, not only decoherence between the various histories of the family,also what these authors call ‘‘inertia’’ to recover classicpredictability!. See also Chap. 16 of Ref. 170 for a discusion of how classical determinism is restored, in a weak vsion that ensures perfect correlations between the valuequasiclassical observables at different times~of course, thereis no question of fundamental determinism in this conte!.The history point of view undoubtedly has many attract


allalp-is-

l-

le

-nm-in

ero-nt

a-

y

e

y

s-

hatas

alesstesi-llye

e

yf.is-s-nsnt

utl-r-of

features, and seems to be particularly clear and easy toat least as long as one limits oneself to one single consisfamily of histories.

How does the history interpretation deal with the existenof several consistent families? They are alla priori equallyvalid, but they will obviously lead to totally different descriptions of the evolution of the same physical system; tis actually the delicate aspect of the interpretation~we willcome back to it in the next section!. The answer of the his-tory interpretation to the question is perfectly clear: Differeconsistent families are to be considered as mutually exsive ~except, of course, in very particular cases wheretwo families can be embedded into a single large consisfamily!; all families may be used in a logical reasoning, bnever combined together. In other words: The physicisfree to choose any point of view in order to describe tevolution of the system and to ascribe properties to the stem; in a second independent step, another consistent famay also be chosen in order to develop other logical conerations within this different frame; but it would be totalmeaningless~logically inconsistent! to combine consider-ations arising from the two frames. This a very importafundamental rule that must be constantly kept in mind whone uses this interpretation. We refer the reader to Ref.for a detailed and systematic discussion of how to reaconsistently in the presence of disparate families, and to R174 for simple examples of incompatible families of histries ~photon hitting a beam splitter, Sec. II! and the discus-sion of quantum incompatibility~Sec. V!; various classicalanalogies are offered for this incompatibility, includingtwo-dimensional representation of a three-dimensional obby a draftsman, who can choose many points of viewmake a drawing, but can certainly not take several atsame time—otherwise the projection would become incsistent.

4. Comparison with other interpretations

In the history interpretation, as we have already seen, this no need to invoke conscious observers, measuremenparatuses, etc.; the system has properties in itself, as innonorthodox interpretations that we discussed before~con-sidering that the correlation interpretation is orthodox!. Astriking feature of the history interpretation, when comparto the others, is the enormous flexibility that exists for tselection of the point of view~family! that can be chosen fodescribing the system, since all the timest1 ,t2 ,... arearbi-trary ~actually their number is also arbitrary! and, for each ofthem, many different projectorsP may be introduced. Onemay even wonder if the interpretation is sufficiently specifiand if this very large number of families of histories is notproblem. This question will come naturally in a comparisbetween the history interpretation and the other interpretions that we have already discussed.

First, what is the exact relation between the history intpretation and the orthodox theory? The relation is certaivery close, but several concepts are expressed in a mprecise way. For instance, complementarity stands inCopenhagen interpretation as a general, almost philosocal, principle. In the history interpretation, it is relatedmathematical conditions, such as consistency conditioalso, every projector cannot be more precise than the protor over a single quantum stateuw&, which is itself obviouslysubject to the uncertainty relations because of the very st

688F. Laloe

t,roteunte

tomonglo

roemthofel

to

oao

ofiffei

naethpr

ie

...

eoxuc

taiooeyeeo

hariye-amud.ind. Fabb

aler

in-le

com-the

ele-

lyal-

onoryts-e

nebyace

aadde-ent.er-Onare

anbe-cs.t-

trat-ousin-enthe-

ghtap-roonr-tlyd toton ifat

heby

ey inf ade-theerm-

ofees-pic

ture of the space of states. Of course, considerations oncompatible measurement devices may still be made buthe Bohrian distinction between the macroscopic and micscopic worlds, they lose some of their fundamental characIn the same vein, the history interpretation allows a quanttheory of the universe@compare for instance with quotatio~v! at the end of Sec. II#; we do not have to worry aboudividing the universe into observed systems and observThe bigger difference between the orthodox and the hisinterpretations is probably in the way they describe the tievolution of a physical system. In the usual interpretatiwe have two different postulates for the evolution of a sinentity, the state vector, which may sometimes create cflicts; in the history interpretation, the continuous Sch¨-dinger evolution and the random evolution of the systamong histories are put at very different levels, so thatconflict is much less violent. Actually, in the history pointview, the Schro¨dinger evolution plays a role only at the levof the initial definition of consistent families~through theevolution operators that appear in the Heisenberg opera!and in the calculation of the probabilityP(H); the real timeevolution takes place between the timest i and t i 11 and ispurely stochastic. In a sense, there is a kind of inversionpriorities, since it is now the nondeterminist evolution thbecomes the major source of evolution, while in the orthdox point of view it is rather the deterministic evolutionan isolated system. Nevertheless, and despite these dences, the decoherent history interpretation remains vmuch in the spirit of the orthodox interpretation; indeed,has been described as an ‘‘extension of the Copenhageterpretation,’’ or as ‘‘a way to emphasize the internal logicconsistency of the notion of complementarity.’’ On the othhand, Gell-Mann takes a more general point of view onhistory interpretation which makes the Copenhagen intertation just ‘‘a special case of a more general interpretationterms of the decoherent histories of the universe. The Cophagen interpretation is too special to be fundamental~Ref. 175!.

What about the ‘‘correlation interpretation?’’ In a sensthis minimal interpretation is contained in both the orthodinterpretation~from which some elements such as the redtion of the state vector have been removed! and in the historyinterpretation. Physicists favoring the correlation interpretion would probably argue that adding a physical discussin terms of histories to their mathematical calculationprobabilities does not add much to their point of view: Thare happy with the calculation of correlations and do not fthe need for making statements on the evolution of the prerties of the system itself. Moreover, they might add tthey wish to insert whatever projectors correspond to a seof measurements in~37!, and not worry about consistencconditions: In the history interpretation, for arbitrary squences of measurements, one would get inconsistent flies for the isolated physical system, and one has to inclthe measurement apparatuses to restore consistencyhave already remarked in Sec. VI A that the correlationterpretation allows a large flexibility concerning the bounary between the measured system and the environmentthese physicists, the history description appears probmore as an interesting possibility than as a necessity;there is no contradiction either.

Are there also similarities with theories with additionvariables? To some extent, yes. Within a given family, thare many histories corresponding to the same Schro¨dinger


in-as-r.

m

rs.rye,

en-

e

rs

ft-

er-rytin-lree-nn-’’

,

-

-nf

lp-tes

i-e

We--orlyut

e

evolution and, for each history, we have seen that moreformation on the evolution of physical reality is availabthan through the state vector~or wave function! only. Underthese conditions, the state vector can be seen as a nonplete description of reality, and one may even argue thathistories themselves constitute additional variables~but theywould then be family dependent, and therefore not EPRments of reality, as we discuss later!. In a sense, historiesprovide a kind of intermediate view between an infiniteprecise Bohmian trajectory for a position and a very delocized wave function. In the Bohm theory, the wave functipilots the position of the particles; in the decoherent histinterpretation, the propagation of the wave function pilorather the definition of histories~through a consistency condition! as well as a calculation of probabilities, but not thevolution between timest i andt i 11 , which is supposed to befundamentally random. Now, of course, if one wished, ocould make the two sorts of theories even more similarassuming the existence of a well-defined point in the spof histories; this point would then be defined as moving incompletely different space from the Bohm theory—insteof the configuration space, it would move in the spacefined by the family, and thus be defined as family dependIn this way, the history interpretation could be made detministic if, for some reason, this was considered useful.many other aspects, the theories with additional variablesvery different from the history interpretation and we cprobably conclude this comparison by stating that theylong to rather different point of view on quantum mechani

Finally, what is the comparison with theories incorporaing additional nonlinear terms in the Schro¨dinger evolution?In a sense, they correspond to a completely opposite segy: They introduce into one single equation the continuevolution of the state vector as well as a nonlinear determistic mechanism simulating the wave packet reduction whneeded; the history interpretation puts on different levelscontinuous Schro¨dinger evolution and a fundamentally random selection of history selection by the system. One miventure to say that the modified nonlinear dynamicsproach is an extension of the purely wave program of Sch¨-dinger, while the history interpretation is a modern versiof the ideas put forward by Bohr. Another important diffeence is that a theory with modified dynamics is not stricequivalent to usual quantum mechanics, and could leaexperimental tests, while the history interpretation is builtreproduce exactly the same predictions in all cases—eveit can sometimes provide a convenient point of view thallows one to grasp its content more conveniently~Ref. 130!.

5. A profusion of points of view; discussion

We finally come back to a discussion of the impact of tprofusion of possible points of view, which are providedall the families that satisfy the consistency condition. Whave already remarked that there is, by far, no single wathis interpretation to describe the evolution of properties ophysical system—for instance, all the complementaryscriptions of the Copenhagen interpretation appear atsame level. This is indeed a large flexibility, much largthan in classical physics, and much larger than in the Bohian theory for instance. Is the ‘‘no combination of pointsview’’ fundamental rule really sufficient to ensure that ththeory is completely satisfactory? The answer to this qution is not so clear for several reasons. First, for macrosco

689F. Laloe

roh

fathn

da

ubdio,ththrro

engodoholivbat

uci-ohit

wt

apheo

seu

eafe

ererieisin

etft nsiuroeteestohiro

n-fo

go:rthetionef.de-er-

eta-her

in

seer aofeta-ng

dictthe

notn-

tor-

hatthe

tt,i-r-

i-e-pleup-

doxiveined,All

ontois

tedand

oned toapt.s a

ex-t. Ita-asthisell

theenon-

systems, one would like an ideal theory to naturally intduce a restriction to sets corresponding to quasiclassicaltories; unfortunately, the number of consistent sets is inmuch too large to have this property, Ref. 176. This isreason why more restrictive criteria for mathematically idetifying the relevant sets are~or have been! proposed, but nocomplete solution or consensus has yet been found; thetailed physical consequences of consistency conditionsstill being explored, and actually provide an interesting sject of research. Moreover, the paradoxes that we havecussed above are not all solved by the history interpretatSome of them are, for instance the Wigner friend paradoxthe extent where no reference to observers is made ininterpretation. But some others are not really solved, andinterpretation just leads to a reformulation in a different fomalism and vocabulary. Let us for instance take the Sch¨-dinger cat paradox, which initially arose from the absenceany ingredient in the Schro¨dinger equation for the emergencof single macroscopic result—in other words, for excludiimpossible macroscopic superpositions of an isolated, nobserved, system. In the history interpretation, the paratransposes in terms of choice of families of histories: Tproblem is that there is no way to eliminate the familieshistories where the cat is at the same time dead and aactually, most families that are mathematically acceptathrough the consistency condition contain projectors on mroscopic superpositions, and nevertheless have exactlysame status as the families that do not. One would mprefer to have a ‘‘superconsistency’’ rule that would elimnate these superpositions; this would really solve the prlem, but such a rule does not exist for the moment. At tstage, one can then do two things: either consider thatchoice of sensible histories and reasonable points of viea matter of good sense—a case in which one returns tousual situation in the traditional interpretation, where theplication of the postulate of wave packet is also left to tgood taste of the physicist; or invoke decoherence and cpling to the external world in order to eliminate all theunwanted families—a case in which one returns to the ussituation where, conceptually, it is impossible to ascribe rsonable physical properties to a closed system without rering to the external world and interactions with it,48 whichagain opens the door to the Wigner friend paradox, etc.

Finally one may note that, in the decoherent history intpretation, there is no attempt to follow ‘‘in real time’’ thevolution of the physical system; one speaks only of histothat are seen as complete, ‘‘closed in time,’’ almost as htories of the past in a sense. Basic questions that weretially at the origin of the introduction of the wave packpostulate, such as ‘‘how to describe the physical reality ospin that has already undergone a first measurement buyet a second,’’ are not easily answered. In fact, the contency condition of the whole history depends on the futchoice of the observable that will be measured, which dnot make the discussion simpler than in the traditional inpretation, but maybe even more complicated since its vlogical frame is now under discussion. What about a seriemeasurements which may be, or may not be, continued infuture, depending on a future decision? As for the EPR crelation experiments, they can be reanalyzed within thetory interpretation formalism, Ref. 177~see also Ref. 101 foa discussion of the Hardy impossibilities and the notion‘‘consistent contrafactuality’’!; nevertheless, at a fundametal level, the EPR reasoning still has to be dismissed


-is-cte-

e-re-is-n.toise

-of

n-x

efe;lec-heh

b-sheishe-

u-

al-r-

-

s-i-

aot

s-es

r-ryofher-s-

f

r

exactly the same reason that Bohr invoked already long aIt introduces the EPR notion of ‘‘elements of reality,’’ ocounterfactual arguments, that are not more valid withinhistory interpretation than in the Copenhagen interpreta~see for instance Sec. V of Ref. 177 or the first letter in R175!. We are then brought back to almost the same oldbate, with no fundamentally new element. We have nevtheless already remarked that, like the correlation interprtion, the history interpretation may be supplemented by otingredients, such as the Everett interpretation49 or, at theother extreme, EPR or deterministic ingredients, a casewhich the discussion would of course become different.

For a more detailed discussion of this interpretation,the references given at the beginning of this section; fodiscussion of the relation with decoherence, the notion‘‘preferred ~pointer! bases,’’ and classical predictability, seRef. 176; for a critique of the decoherent history interpretion, see for instance Ref. 178, where it is argued amoothers that consistency conditions are not sufficient to prethe persistence of quasiclassicality, even at large scales inUniverse; see also Ref. 179, which claims that they aresufficient either for a derivation of the validity of the Copehagen interpretation in the future; but see also the replythis critique by Griffiths in Ref. 174. Finally, another refeence is a recent article in Physics Today~Ref. 16! that con-tains a discussion of the history interpretation in terms tstimulated interesting reactions from the proponents ofinterpretation~Ref. 175!.

E. Everett interpretation

A now famous point of view is that proposed by Everewho named it ‘‘relative state interpretation’’—but in its varous forms it is sometimes also called ‘‘many-worlds intepretation,’’ or ‘‘branching universe interpretation’’~the word‘‘branching’’ refers actually to the state vector of the unverse!. In this interpretation, any possible contradiction btween the two evolution postulates is canceled by a simbut efficient method: The second postulate is merely spressed!

In the Everett interpretation~Ref. 180!, the Schro¨dingerequation is taken even more seriously than in the orthointerpretation. Instead of trying to explain how successsequences of well-defined measurement results are obtaone merely considers that single results never emerge:possibilities are in fact realized at the same time! The VNeumann chain is never broken, and its tree is left freedevelop its branch ad infinitum. The basic remark of thinterpretation is that, for a composite system of correlasubsystems~observed system, measurement apparatus,observer, all considered after a measurement!, ‘‘there doesnot exist anything like a single state for one subsystem...can arbitrarily choose a state for one subsystem and be lethe relative state for the remainder’’—this is actually justdescription of quantum entanglement, a well-known conceBut, now, the novelty is that the observer is considered apurely physical system, to be treated within the theoryactly on the same footing as the rest of the environmencan then be modeled by an automatically functioning mchine, coupled to the recording devices and registering psensory data, as well as its own machine configurations. Tleads Everett to the idea that ‘‘current sensory data, as was machine configuration, is immediately recorded inmemory, so that all the actions of the machine at a givinstant can be considered as functions of the memory c

690F. Laloe

boro

eoso

erne

itsurhehgminta-ofnsheke

Inrnioerlctheheio

nd, tnthueoatothtaeanwuthe

pu, bucytee

s,o

thcth

torstic

thery.t isettytons,ho

tro-

ore

um

chin-gle-the

ults,ep-herm-af a

ot,uce

re-nspe-

nic

oths ofn-om-asre-

be

NAdllyt of

ol-ple

ects,er-n-

aywillwent

hellel!

tents only... .’’ Similarly, all relevant experience that the oserver keeps from the past is also contained in this memFrom this Everett concludes that ‘‘there is no single statethe observer;...with each succeeding observation~or interac-tion!, the observer state branches into a number of differstates... . All branches exist simultaneously in the superption after any sequence of observations.’’ Under these cditions, the emergence of well-defined results from expments is not considered as a reality, but just as a delusiothe mind of the observer. What the physical system dotogether with the environment, is to constantly ramifystate vector into all branches corresponding to all measment results, without ever selecting one of these brancThe observer is also part of this ramification process, tnevertheless has properties which prevent him/her to brinhis/her mind the perception of several of them at the satime. Indeed, each ‘‘component of the observer’’ remacompletely unaware of all the others, as well as of the svectors that are associated with them~hence the name ‘‘relative state interpretation’’!. The delusion of the emergencea single result in any experiment then appears as a coquence of the limitations of the human mind: In fact, tprocess that we call ‘‘quantum measurement’’ never taplace!

How is an EPRB experiment seen in this point of view?the Bohmian interpretation we emphasized the role of SteGerlach analyzers, in the nonlinear evolution interpretatthat of the detectors and decoherence; here we have tophasize the role of the correlations with the external woon the mind of the two human observers. The state vewill actually develop its Von Neumann chain through tanalyzers and the detectors and, at some point, include tobservers whose brain will become part of the superpositFor each choice of the settingsa andb, four branches of thestate vector will coexist, containing observers whose miare aware of the result associated with each branch. Sochoice ofa has a distant influence on the mind of the secoobserver, through the definition of the relevant basis forVon Neumann chain, and nonlocality is obtained as a res

It is sometimes said that ‘‘what is most difficult in thEverett interpretation is to understand exactly what one dnot understand.’’ Indeed, it may look simple and attractivefirst sight, but turns out to be as difficult to defend asattack~see nevertheless Sec. 3 of Ref. 181, where the auconsiders the theory as ambiguous because dynamical sity conditions are not considered!. The question is, to somextent, what one should expect from a physical theory,what it should explain. Does it have to explain in detail howe perceive results of experiments, and if so of what natshould such an explanation be? What is clear, anyway, isthe whole point of view is exactly opposite to that of thproponents of the additional variables: The emphasis isnot on the physical properties of the systems themselveson the effects that they produce on our minds. Notions sas perception~Ref. 180 speaks of ‘‘trajectory of the memorconfiguration’’! and psychology become part of the debaBut it remains true that the Everett interpretation solvbeautifully all difficulties related to Bohrian dichotomieand makes the theory at the same time simpler and mpleasant aesthetically. Since the human population of earmade of billions of individuals, and presumably since eaof them is busy making quantum measurements all of


-y.f

nti-

n-i-ofs,

e-s.

attoe

ste

e-

s

–nm-dor

sen.

shedelt.

est

orbil-

d

reat

t,uth

.s

reis

he

time without even knowing it, should we see the state vecof the universe as constantly branching at a really fantarate?

VII. CONCLUSION

Quantum mechanics is, with relativity, the essence ofbig conceptual revolution of the physics of the 20th centuNow, do we really understand quantum mechanics? Iprobably safe to say that we understand its machinery prwell; in other words, we know how to use its formalismmake predictions in an extremely large number of situatioeven in cases that may be very intricate. Heinrich Hertz, wplayed such a crucial role in the understanding of elecmagnetic waves in the 19th century~Hertzian waves!, re-marked that, sometimes, the equations in physics are ‘‘mintelligent than the person who invented them’’~Ref. 182!.The remark certainly applies to the equations of quantmechanics, in particular to the Schro¨dinger equation, or tothe superposition principle: They contain probably mumore substance that any of their inventors thought, forstance in terms of unexpected types of correlations, entanment, etc. It is astonishing to see that, in all known cases,equations have always predicted exactly the correct reseven when they looked completely counterintuitive. Conctually, the situation is less clear. One major issue is whetor not the present form theory of quantum mechanics is coplete. If it is, it will never be possible in the future to givemore precise description of the physical properties osingle particle than its wave function~or of two particles, forinstance, in an EPR-type experiment!; this is the position ofthe proponents of the Copenhagen interpretation. If it is nfuture generations may be able to do better and to introdsome kind of description that is more accurate.

We have shown why the EPR argument is similar to Ggor Mendel’s reasoning, which led him from observatioperformed between 1854 and 1863 to the discovery of scific factors, the genes~the word appeared only later, i1909!, which turned out to be associated with microscopobjects hidden inside the plants that he studied. In bcases, one infers the existence of microscopic ‘‘elementreality’’ from the results of macroscopic observations. Medel could derive rules obeyed by the genes, when they cbine in a new generation of plants, but at his time it wtotally impossible to have any precise idea of what theyally could be at a microscopic level~or actually even if theywere microscopic objects, or macroscopic but too small toseen with the techniques available at that time!. It took al-most a century before O. T. Avery and colleagues~1944!showed that the objects in question were contained in Dmolecules; later~1953!, F. Crick and J. Watson illustratehow subtle the microscopic structure of the object actuawas, since genes corresponded to subtle arrangemennucleic bases hidden inside the double helix of DNA mecules. We now know that, in a sense, rather than simmicroscopic objects, the genes are arrangements of objand that all the biological machinery that reads them is ctainly far beyond anything that could be conceived at Medel’s time. Similarly, if quantum mechanics is one dsupplemented with additional variables, these variablesnot be some trivial extension of the other variables thatalready have in physics, but variables of a very differenature. But, of course, this is only a possibility, since thistories of biology and physics are not necessarily para

691F. Laloe

inhitarathc

i-dh

blrebeite

ll iec

-ta

i

toli-nne-uo

thesSt

ulsongoinfollyetemll ao

rilyextyhfuef

veason-es-

se

ionsums, inult,ay

it isandtumre-reies:an-ale!thatme

on-orx-

thatis

heies.as-

ef-

oftionovereof

intoics,thethat-like

ionjus-

letom

Anyway, the discussion of additional variables leads toteresting questions, which we have tried to illustrate in tarticle by a brief description of several possible interpretions of quantum mechanics that have been or are still pposed; some introduce additional variables that indeed hvery special properties, others do not, but in any casetheory, at some stage, contains features that are reminisof these difficulties.

A natural comparison is with special relativity, since nether quantum mechanics nor relativity is intuitive; indeeexperience shows that both, initially, require a lot of thougfrom each of us before they become intellectually acceptaBut the similarity stops here: While it is true that, the moone thinks about relativity, the more understandable itcomes~at some point, one even gets the feeling that relativis actually a logical necessity!!, one can hardly say the samthing about quantum mechanics. Nevertheless, among atellectual constructions of the human mind, quantum mchanics may be the most successful of all theories sindespite all efforts of physicists to find its limits of validity~asthey do for all physical theories!, and many sorts of speculation, no one for the moment has yet been able to obclear evidence that they even exist. The future will tell usthis is the case; surprises are always possible!

ACKNOWLEDGMENTS

The first version of this text was written during a visitthe Institute of Theoretical Physics of the University of Cafornia at Santa Barbara, as a side activity during a sessioBose–Einstein condensation, making profit of the preseof A. Leggett and sharing an office with W. Zurek; the rsearch was supported in part by the National Science Fodation under Grant No. PHY94-07194. The final versionthe text was made during a visit to the Lorentz Center ofUniversity of Leiden, as a side activity during another ssion on Bose–Einstein condensation, two years later;Stenholm was kind enough to read the whole article andprovide useful advice and comments. The intellectual stimlation of these two visits was wonderful! The author is avery grateful to William Mullin, Philippe Grangier, JeaDalibard, S. Goldstein, Serge Reynaud, and Olivier Darrifor useful comment and advice; they stimulated the rewritof various parts of this text, sometimes of whole sections,better clarity. Abner Shimony was kind enough to carefuread several versions of this manuscript; the author is vgrateful for all useful suggestions that he made at every sMany thanks are also due to Robert Griffiths for his coments on the section on the history interpretation, as weto P. Pearle and G. Ghirardi concerning the section on nlinear Schro¨dinger dynamics. It is hoped, but not necessatrue, that all these colleagues will agree with the present tor at least most of it; if not, the scientific responsibilishould be considered as entirely that of the present autFinally, an anonymous referee made two long, careand interesting reports containing several especially ussuggestions, which were taken with gratitude.

LKB ~Laboratoire Kastler Brossel! is a ‘‘Unite MixteCNRS,’’ UMR 8552.


-s-

o-vee

ent

,te.

-y

n--e,

inf

once

n-fe-igto-

lgr

ryp.-s

n-

t,

or.lul

APPENDIX A: AN ATTEMPT TO CONSTRUCT A‘‘SEPARABLE’’ QUANTUM THEORY„NONDETERMINISTIC BUT LOCAL THEORY …

We come back to the discussion of Sec. III B but now giup botany; in this Appendix we consider a physicist who hcompletely assimilated the rules of quantum mechanics ccerning nondeterminism, but who is skeptical about thesential character of nonlocality in this theory~or nonsepara-bility; for a detailed discussion of the meaning of theterms, see, for instance, Refs. 21 and 34!. So, this physicistthinks that, if measurements are performed in remote regof space, it is more natural to apply the rules of quantmechanics separately in these two regions. In other wordorder to calculate the probability of any measurement reshe/she will apply the rules of quantum mechanics in a wthat is perfectly correct locally; the method assumes thatpossible to reason separately in the two regions of space,therefore ignores the nonseparable character of quanevents~quantum events may actually involve both spacegions at the same time!. Let us take an extreme case, whethe two measurements take place in two different galaxOur physicist would be prepared to apply quantum mechics to the scale of a galaxy, but not at an intergalactic sc

How will he/she then treat the measurement processtakes place in the first galaxy? It is very natural to assuthat the spin that it contains is described by a state vector~orby a density operator, it makes no difference for our reasing here! that may be used to apply the orthodox formula fobtaining the probabilities of each possible result. If our eperimenter is a good scientist, he/she will realize at onceit is not a good idea to assume that the two-spin systemdescribed by a tensor product of states~or of density opera-tors!; this would never lead to any correlation between tresults of measurements performed in the two galaxTherefore, in order to introduce correlations, he/she willsume that the states in question~or the density operators! arerandom mathematical objects, which fluctuate under thefect of the conditions of emission of the particles~for in-stance!. The method is clear: For any possible conditionthe emission, one performs an orthodox quantum calculain each region of space, and then takes an average valuethe conditions in question. After all, this is nothing but thuniversal method for calculating correlations in all the restphysics! We note in passing that this approach takesaccount the indeterministic character of quantum mechanbut introduces a notion of space separability in the line ofEPR reasoning. Our physicist may for instance assumethe two measurement events are separated by a spaceinterval in the sense of relativity, so that no causal relatcan relate them in any circumstance; this seems to fullytify an independent calculation of both phenomena.

Even if this is elementary, and for the sake of clarity,us give the details of this calculation. The fluctuating randvariable that introduces the correlations is calledl, and thedensity operator of the first spinr1(l); for a direction ofmeasurement defined by the unit vectora, the eigenstate ofthe measurement corresponding to result11 is denotedu1/a&. The probability for obtaining result1 if the firstmeasurement is made along directiona is then written as:

P1~a,l!5^1/aur1~l!u1/a&. ~44!

In the same way, one writes the probability for the result21in the form:

692F. Laloe

twe

e-lts

ae

thge

2

na

sie

nearby

B

eena

tureodwteTeicityell

ker

ctre

o

re-

rumer

o-ella

en-isthe

nsea-

areate

ut

torctre

as-

P2~a,l!5^2/aur1~l!u2/a&. ~45!

If, instead of directiona, another different directiona8 ischosen, the calculations remain the same and lead tofunctionsP6(a8,l). As for measurements performed in thsecond region of space, they provide two functionsP6(b,l)andP6(b,l).

We now calculate the number which appears in the Btheorem~BCHSH inequality!, namely the linear combination, as in~6!, of four average values of products of resuassociated with the couples of orientations (a,b),(a,b8),(a8,b),(a8,b8). Since we have assumed that results areways61, the average value depends only on the differenc

A~l!5P1~a,l!2P2~a,l! ~46!

or

A8~l!5P1~a8,l!2P2~a8,l! ~47!

~with similar notation for the measurements performed inother region of space! and can be written as the averavalue overl of

A~l!B~l!1A~l!B8~l!2A8~l!B~l!1A8~l!B8~l!.~48!

We are now almost back to the calculation of Sec. IV Awith a little difference nevertheless: TheA’s andB’s are nowdefined as probability differences so that their values arenecessarily61. It is nonetheless easy to see that they arebetween11 and21, whatever the value ofl is. Let us for amoment considerl, A, andA8 as fixed, keeping onlyB andB8 as variables; in the space of these variables, expres~48! corresponds to a plane surface which, at the four cornof the squareB561, B8561, takes values62A or 62A8,which are between62; at the center of the square, the plagoes through the origin. By linear interpolation, it is clethat, within the inside of the square, the function given~48! also remains bounded between62; finally, its averagevalue has the same property. Once more we find that thetheorem holds in a large variety of contexts!

Since we know that quantum mechanics as well as expments violate the Bell inequality, one may wonder what wwrong in the approach of our physicist; after all, his/her resoning is based on the use of the usual formalism of quanmechanics. What caused the error was the insistence of ting the measurements as separable events, while orthquantum mechanics requires us to consider the whole tspin system as a single, nonseparable, system; in this sysno attempt should be made to distinguish subsystems.only correct reasoning uses only state vectors/density optors that describe this whole system in one mathematobject. This example illustrates how it is really separabiland/or locality which are at stake in a violation of the Binequalities, not determinism.

It is actually instructive, as a point of comparison, to mathe calculation of standard quantum mechanics as similapossible to the reasoning that led to the inequality~48!. Forthis purpose, we notice that any density operatorr of thewhole system belongs to a space that is the tensor produthe corresponding spaces for individual systems; thereforcan always be expanded as:

r5(n,p

cn,p@rn~1! ^ rp~2!#. ~49!


o

ll

l-s:

e

,

otll

onrs

ell

ri-t-mat-oxo-m,

hera-al

as

of

From this, one can obtain the probability of obtaining twresults11 along directionsa andb as:

P11~a,b!5(n,p

cn,p^1/aurnu1/a&^1/burpu1/b& ~50!

~probabilities corresponding to the other combinations ofsults are obtained in the same way!. The right-hand side ofthis equation is not completely different from the sum ovelthat was used above; actually it is very similar, since the sover the indicesn andp plays the same role as the sum ovthe different values ofl. In fact, if all cn,p’s were real posi-tive numbers, and if all operatorsrn andrp were positive~orsemipositive! operators, nothing would prevent us from ding exactly the same reasoning again and deriving the Binequality; in other words, any combined system that isstatistical mixture~which implies positive coefficients! ofuncorrelated states satisfies the Bell inequalities. But, in geral, the positivity conditions are not fulfilled, and thisprecisely why the quantum mechanical results can violateinequalities.

APPENDIX B: MAXIMAL PROBABILITY FOR AHARDY STATE

In this Appendix we give more details on the calculatioof Sec. IV B; the two-particle state corresponding to the msurement considered in~i! is the tensor product of ket~9! byits correspondent for the second spin:

cos2 uu1,1&1sinu cosu@ u1,2&1u2,1&] 1sin2 uu2,2&,~51!

which has the following scalar product with ket~14!:

cos2 u sinu22 sinu cos2 u52sinu cos2 u. ~52!

The requested probability is obtained by dividing the squof this expression by the square of the norm of the stvector:

P5sin2 u cos4 u

2 cos2 u1sin2 u5

sin2 u~12sin2 u!2

22sin2 u. ~53!

A plot of this function shows that it has a maximum of abo0.09.

APPENDIX C: PROOF OF RELATIONS „17… AND„18…

Let us start with the ket:

uC&5u1,1,1&1hu2,2,2&, ~54!

where

h561. ~55!

We wish to calculate the effect of the product operas1xs2ys3y on this ket. Since every operator in the producommutes with the two others, the order in which they aapplied is irrelevant; let us then begin with the operatorsociated with the first spin:

s11uC&52hu1,2,2&,~56!s12uC&52u2,1,1&,

which provides

s1xuC&5uC8&5hu1,2,2&1u2,1,1&. ~57!

693F. Laloe

ue

pi

ecoruly

blets

rsorretay

omw

saoro

ofu

ergeoneesetionu-ty,fect

r tony

ucesus-

eralnlyre-thesed

toentor-alland

,rewe

the

s

dyicheri-um

es a

ocalary

ver

re-

op-anyt a

ef

here

t:ch

ob-

For the second spin,

s21uC8&52hu1,1,2&,~58!s22uC8&52u2,2,1&,

so that

s2yuC8&5uC9&51

i~hu1,1,2&2u2,2,1&). ~59!

Finally, the third spin gives

s31uC9&522ihu1,1,1&,~60!s32uC9&512i u2,2,2&,

which leads to

s3yuC9&52hu1,1,1&2u2,2,2&52huC& ~61!

~sinceh251!. Indeed, we find thatuC& is an eigenstate of theproduct of the three spin operatorss1xs2ys3y , with eigen-value 2h. By symmetry, it is obvious that the same is trfor the product operatorss1ys2xs3y ands1ys2ys3x .

Let us now calculate the effect of operators1xs2xs3x onuC&; from ~58! we get

s2xuC8&5uC-&5~hu1,1,2&1u2,2,1&) ~62!

so that

s31uC-&52hu1,1,1&,~63!s32uC-&52u2,2,2&,

and, finally,

s3xuC-&5hu1,1,1&1u2,2,2&5huC&. ~64!

The change of sign between~61! and ~64! may easily beunderstood in terms of simple properties of the Pauli soperators~anticommutation and square equal to one!.

APPENDIX D: IMPOSSIBILITY OF SUPERLUMINALCOMMUNICATION AND OF CLONINGQUANTUM STATES

In EPR schemes, applying the reduction postulate projthe second particle instantaneously onto an eigenstate csponding to the same quantization axis as the first measment. If it were possible to determine this state completesuperluminal communication would become accessiFrom this state, the second experimenter could calculatedirection of the quantization axis to which it correspondand rapidly know what direction was chosen by the fiexperimenter,50 without any special effect of the distance, finstance even if the experimenters are in two differentmote galaxies. This, obviously, could be used as a sortelegraph, completely free of any relativistic minimum del~proportional to the distance covered!. The impossibility forsuperluminal communications therefore relies on the impsibility of a complete determination of a quantum state froa single realization of this state. Such a realization alloonly one single measurement, which~almost always! per-turbs the state, so that a second measurement on thestate is not feasible; there is not, and by far, sufficient infmation in the first measurement for a full determinationthe quantum state—see the discussion in Sec. V D.

Now, suppose for a moment that a perfect ‘‘cloning’’quantum states could be performed—more precisely the mtiple reproduction~with many particles! of the unknown state


n

tsre-re-,:

he,t

-or

s-

s

me-f

l-

of a single particle.51 Applying the cloning process to thsecond particle of an EPR pair, one could then make a lanumber of perfect copies of its state; in a second step,could perform a series of measurements on each of thcopies, and progressively determine the state in queswith arbitrary accuracy. In this way, the possibility for sperluminal communication would be restored! But, in realiquantum mechanics does not allow either for such a perreproduction of quantum states~Refs. 108 and 109!; for in-stance, if one envisages using stimulated emission in ordeclone the state of polarization of one single photon into macopies, the presence of spontaneous emission introdnoise in the process and prevents perfect copying. A discsion of multiparticle cloning is given in Ref. 110.

This, nevertheless, does not completely solve the genquestion: Even without cloning quantum states, that is owith the information that is available in one single measument in each region of space, it is not so obvious thatinstantaneous reduction of the wave packet cannot be ufor superluminal communication. After all, it is possiblerepeat the experiment many times, with many independpairs of correlated particles, and to try to extract some infmation from the statistical properties of the results ofmeasurements. The EPR correlations are very specialexhibit such completely unexpected properties~e.g., viola-tions of the Bell inequalities!! Why not imagine that, byusing or generalizing EPR schemes~more than two systemsdelocalized systems, etc.!, one could invent schemes whesuperluminal communication becomes possible? Hereshow why such schemes do not exist; we will sketchgeneral impossibility proof in the case of two particles~ortwo regions of space!, but the generalization to more systemin several different regions of space is straightforward.

Suppose that, initially, the two remote observers alreapossess a collection of pairs of correlated particles, whhave propagated to their remote galaxies before the expment starts. Each pair is in an arbitrary state of quantentanglement, and we describe it with a density operatorr ina completely general way. The first observer then choossettinga or, more generally, any local observableOA to mea-sure; the second observer is equally free to choose any lobservableOB , and may use as many particles as necessto measure the frequency of occurrence of each result~i.e.,probabilities!; the question is whether the second obsercan extract some information onOA from any statisticalproperty of the observed results. The impossibility prooflies on the fact that all operators~observables! correspondingto one of the two subsystems always commute with allerators corresponding to the other; consequently, forchoice of the operators, it is always possible to construccommon eigenbasis$uwk ,u j&% in the space of states of thtwo-particle system, where theuwk& ’s are the eigenstates oOA and theuu j& ’s are the eigenstates ofOB . We can thencalculate the probability of sequences of measurement wthe first operator obtains resultAm ~corresponding, if thiseigenvalue is degenerate, to some rangeDm for the indexk!and the second resultBn ~corresponding to rangeDn forindex j!. But, what we are interested in is slightly differenThe probability that the second observer will obtain earesult Bn after a measurement performed by the otherserver, independent of the resultAm , since there is no way to

694F. Laloe

seo

ngo

ar

tearth

.

e

ckemsn-yonrti

e

ram

cehaeme

toerhioiso

reot

ith

rsf

areialent

e oforsbeentto

or-de-er.

tiv-

theu-

atver-tiestheua-pro-er-the

eenatingtheemally

lo-

r atianof

osi-s of

n-em-of

isede atid

hisonles

-rdil-of

di-knotthelds

have access to this result in the second galaxy; our purpoto prove that this probability is independent of the choicethe operatorOA .

Mathematically, extracting the probabilities concernithe second observer only amounts to summing over all psible resultsAm , with the appropriate weights~probabilities!;this is a classical problem, which leads to the notion of ‘‘ptial trace’’ rB over the variables of the subsystemA. Thisoperator acts only in the space of states of systemB and isdefined by its matrix elements:

^u i urBuu j&5(k

^wk ,u i uruwk ,u j&. ~65!

It contains all information that the second experimenneeds for making predictions, exactly as from any ordindensity operator for an isolated system; for instance,probability of observing resultBn is simply

P~Bn!5TrH (i PDn

uu i&^u i urBJ . ~66!

Equations~65! and ~66! can be derived in different waysOne can for instance use formula~37!, if it has been provedbefore. Otherwise, one can proceed in steps: One firstpandsr in terms of projectors onto its own eigenstatesuC l&,with positive eigenvalues; one then applies the wave pareduction postulate to eachuC l& separately in order to get thprobability of any sequence of results; one finally perforthe sum overl as well as the appropriate sum over indicek~unknown result! and j ~if the observed eigenvalue is degeerate! in order to obtain the ‘‘reduced probabilities’’—bthese words we mean the probabilities relevant to the secobserver, just after the other has performed a measuremeOA , but before it has been possible to communicate thesult to the second by some classical channel. This calculaprovides the above expressions.

From formula~65!, one might get the impression that thpartial trace depends on the choice of the basis$uwk&%, sothat there is some dependence of operatorrB on the choiceof OA . This is a false impression: In fact, a simple algebshows that the sum contained in the partial trace is copletely independent of the basis chosen in the traced spastates; it does not even matter if the first experimenterperformed any experiment or not. Therefore, the secondperimenter receives exactly the same information, copletely independent of the decisions made by the first expmenter; no superluminal communication is possible.

Finally, one could object that it is not indispensablehave one system located in one region of space, the oththe second region, as we have assumed until now; eacthem could perfectly well be delocalized in a superpositof states in different locations. Does the proof hold in thcase? Yes, it does, after some modification. In this case,should now associate the lettersA andB, as well as operatorsOA and OB , not to subsystems as before, but to measuments performed in each region of space. Each relevanterator can then be put between two projectors onto statesare localized either in the first~projectorPA!, or the second~projector PB!, region of space. SincePA and PB are or-thogonal, it is then simple to show that all operators windex A commute with all operators with indexB ~this issimilar, in field theory, to the commutation of field operatothat are outside mutual light cones!; this remains true even i


isf

s-

-

rye

x-

et

s

ndt ofe-on

-ofs

x--

ri-

inof

n

ne

-p-

hat

they act in the space of states of the same particle. Wenow dealing with a generalization of the notion of parttrace, which is no longer related to the existence of differsubsystems~it may actually apply to one particle only!, butto two different sets of operators acting in the same spacstates. If all operators of one set commute with all operatof the second set, the notion of partial trace can indeedtransposed, and it turns out that the final result is independof the operator that was chosen in the first set in ordercalculate the trace. This allows one to prove that the infmation available in one region of space is completely inpendent of the kind of measurement performed in the othIndeed, quantum mechanics is not contradictory with relaity!

APPENDIX E: MANIPULATING AND PREPARINGADDITIONAL VARIABLES

Using the hydrodynamic equations associated withevolution of the wave function, in order to guide the evoltion of the additional variables~positions!, may look like avery natural idea. In other fields of physics, it is known ththe hydrodynamic equations can be obtained by taking aages of microscopic quantities over positions and velociof point-like particles; there is some analogy betweenguiding term and the force term in Landau-type kinetic eqtions, where each particle is subject to an average forceportional to the gradient of the density for instance. Nevtheless, here we are dealing with a single particle, so thatguiding term cannot be associated with interactions betwparticles. Moreover, we also know from the beginning thrather unusual properties must be contained in the guidequations, at least if the idea is to exactly reproducepredictions of usual quantum mechanics: The Bell theorstates that the additional variables have to evolve nonlocin ordinary three-dimensional space~on the other hand, inthe configuration space of the system, they may evolvecally, exactly as for the state vector!. In other words, theadditional variables must be able to influence each othean arbitrary distance in real space. Indeed, in the Bohmequation of motion of the additional variables, the velocitya particle contains an explicit dependence on its own ption, as expected, but also a dependence on the positionall the other particles~assuming that the particles are etangled!. This is not a problem in itself: As mentioned in thmain text, one can consider that making nonlocality copletely explicit in the equations is actually an advantageBohmian mechanics.

But one also has to be careful when this nonlocal termincluded in the equations of motion: Since relativity is bason the idea that it is totally impossible to send a messaga velocity exceeding the velocity of light, one must avofeatures in the theory that would create conflicts with tprinciple. We must distinguish two cases, dependingwhether we consider influences on the additional variabthat are direct~one modifies them ‘‘by hand,’’ in a completely arbitrary way, as for instance the position of a billiaball!, or indirect~applying external fields changes the Hamtonian of the system, and therefore modifies the evolutionthe wave function so that, in turn, the evolution of the adtional variables is affected!. In the latter case, one can checthat the nonlocal Bohmian term creates no problem: It canbe used to transmit instantaneous information throughadditional variables. This is a general result, which ho

695F. Laloe

oroinarte

hes

anticus

tnsibth

edinepo

od

nit

m

ni

ibatf io

rreothemeittaa

ofb

ci-t

ndit

ouoon

isfil

tocy

on-sys-thethers

rec-to

hatwe

ofe-tlyon-us.or-ndur-n-atic

the

e

ve

al-

lved

ly

, and

ns.e ofareof

n-erweof

y

aythecu-esit isde-

simply because the statistical predictions of Bohmian theare equivalent to usual quantum mechanics, which itself dnot allow superluminal communication. But assume forstance that we could manipulate directly the additional vable attached to a particle belonging to an EPR correlapair, in a completely arbitrary way~even at a microscopicscale!, and without changing the wave function; then, t‘‘quantum velocity term’’ acting on the additional variableof the other particle would instantaneously be affected,so would its subsequent position in space; since that parmay be in principle at an arbitrary distance, one couldthis property to send messages at a velocity exceedingvelocity of light. The conclusion is that such manipulatioshould be considered as impossible: The only posssource of evolution of the additional variables has to bewave function dependent term.

If the additional variables cannot be directly manipulatat a microscopic scale, can we then somehow filter themrange of values, as one does for the state vector when thOzcomponent is filtered in a Stern–Gerlach apparatus? Supfor instance that we could, for a particle in an eigenstatethe Oz component of its spin, select the values of the adtional variable that will correspond to a result11 in a futuremeasurement of theOx component; were such a selectiopossible with the help of any physical device, the theory wadditional variables would obviously no longer be copletely equivalent to standard quantum mechanics~this isbecause, within orthodox theory, if a spin 1/2 particle is itially selected into the11 spin state by anOz orientedStern–Gerlach apparatus, it becomes completely impossto make any prediction on the deviation observed later inOx oriented Stern–Gerlach apparatus!. Theories such as thadeveloped in Ref. 4 include this as a possibility; indeed, iis ever demonstrated experimentally, there will be very goreasons to abandon standard quantum theory in favotheories with additional variables! Of course, we cannot pdict the future and conceptual revolutions are always psible, but for the moment it may seem safer to provideadditional variable theories with features that make thequivalent to orthodox theory. In this perspective, it becomnecessary to assume that the additional variables can nebe manipulated directly nor filtered, as opposed to the svector. In other words, the additional variables describeobjective reality, but at a different level from the realitythe field of the wave function, since only the latter caninfluenced directly by human decisions.

APPENDIX F: CONSTRUCTING CONSISTENTFAMILIES OF HISTORIES

This Appendix provides a discussion of the consistencondition ~43!. First, we should mention that other condtions have been proposed and used in the literature; ininitial article on histories~Ref. 15!, a weaker condition in-volving only the cancellation of the real part of the left-haside of ~43! was introduced. For simplicity, here we limourselves to the stronger condition~43!, which is a sufficientbut not necessary condition to the weaker form; it turnsthat, as noted in Ref. 178, it seems more useful in this ctext to introduce selectivity than generality in the definitiof consistent histories.

At first sight, a natural question that comes to mindwhether or not it is easy, or even possible at all, to fulexactly the large number of conditions contained in~43!;


yes-i-d

dleehe

lee

a

sef

i-

h-

-

len

tdof-

s-e

sherten

e

y

he

tn-

l

actually, it has been proposed by Gell-Mann and Hartlegive a fundamental role to families that satisfy consistenconditions in only an approximate way~Ref. 169!, but herewe leave aside this possibility and consider only exact csistency conditions. Let us assume for instance that thetem under study is a particle propagating in free space;various projectors may then define ranges of positions forparticle, playing a role similar to diaphragms or spatial filtein optics that confine an optical beam in the transverse dition. Then the consistency condition will appear as similara noninterference condition for the Huyghens wavelets tare radiated by the inner surface of each diaphragm. Butknow that diffraction is unavoidable in the propagationlight; even if it can be a very small effect when the wavlength is sufficiently short and the diaphragms sufficienbroad, it is never strictly zero. Can we then satisfy the ninterference conditions exactly? The answer is not obvioIt turns out to be yes, but it is necessary to exploit the enmous flexibility that we have in the choice of subspaces aprojectors in a large space of states, and not to limit oselves to projectors over well-defined positions only. To uderstand why, we now briefly sketch one possible systemmethod to construct consistent families of histories.

The simplest method is to guide the construction onstructure of~43!, and to introduce the eigenstatesuwn

0& of thedensity operatorr(t0) ~a Hermitian operator can always b

diagonalized!; let us then define the operatorsP1,j 1(t1) as:

P1,n~ t1!5uwn0&^wn

0u, ~67!

which is equivalent to assuming that their Schro¨dinger coun-terparts P1,j are the projectors over the states that haevolved from theuwn

0& ’s from time t0 to time t1 . Becauser(t0) is of course diagonal in its own basis, this choiceready ensures the presence of a factord j 1 , j 18

on the right-

hand side of~43!. Now, we can also assume that theP2,j 2’s

are defined as the projectors over the states that have evofrom the uwn

0& ’s from time t0 to time t2 , so that a relationsimilar to ~67! is again obtained; this will ensure, not onthe presence of factorsd j 2 , j 28

on the right-hand side of~43!,

but actually also the appearance of a delta functiond j 1 , j 2.

The procedure can be repeated as many times as neededin this way a consistent family is built.

It is nevertheless a very special family, for several reasoThe first is that each projector corresponds to a subspacdimension one only, which corresponds to histories that‘‘maximally accurate;’’ the second is that most historiesthe family have zero probability: In fact, only those withj 1

5 j 25 j 35... are possible, which means that the only radomness occurs at timet1 , and that all subspaces at lattimes are then perfectly determined. The description thatobtain is, in a sense, trivial: Initially, the system is in onethe eigenstates that are contained inr(t0), and then evolvesdeterministically from this initial state.

But it is possible to make the family less singular bgrouping together, for each timet i , several projectors intoone single projector; different associations of projectors mbe used at different times. In this way, the description ofevolution of the state within this family becomes less acrate, but also less trivial since projectors at different timare no longer associated pair by pair. On the other hand,possible to see that this grouping of projectors has not

696F. Laloe

sels

ve

rddies

ly

leaglity

ctio

lidon. 8

timm

comea

ntith,

r i

dn

hee

oml r

tum:

on

-thpf

eaeas

one

.rs

thetwo,

bled forohr’s

s

PR

ennd to

do-hy,

ell

ut

c-

ntsea-surearilydif-

ch as

aulit of

e

airsing:

anythe

s, ofare

k atthe

thealso,ientw’’de-

gu-

airs

ies,

alues

thend;nent

can

stroyed the consistent character of the family; of courother methods for constructing consistent families are apossible.

NOTES „indicated in the text by superscripts…

1In this article, we will not make any distinction between the words ‘‘wafunction’’ and ‘‘state vector.’’

2As we discuss in more detail in Sec. VI B, we prefer to use the wo‘‘additional variables’’ since they are not hidden, but actually appearrectly in the results of measurements; what is actually hidden in ththeories is rather the wave function itself, since it evolves independentthese variables and can never be measured directly.

3It is amusing to contrast the titles of Refs. 8 and 16.4For instance, the nonlocality effects occurring with two correlated particcan be seen as a consequence of the fact that the wave function proplocally, but in a six-dimension space, while the usual definition of locarefers to ordinary space which has three dimensions.

5One should probably mention at this point that quantum mechanicsindeed be formulated in a way which does not involve the configuraspace, but just the ordinary space: the formalism of field operators~some-times called second quantization for historical reasons!. The price to pay,however, is that the wave function~a complex number! is then replaced byan operator, so that any analogy with a classical field is even less va

6Here we just give a simplified discussion; in a more elaborate context,would introduce, for instance, the notion of intersubjectivity, etc., Refs21.

7We implicitly assume that the two observers use the same space–referential; otherwise, one should apply simple mathematical transfortions to go from one state vector to the other. But this has no moreceptual impact than the transformations which allow us, in classicalchanics, to transform positions and conjugate momenta. We shouldthat there is also room in quantum mechanics for classical uncertaiarising from an imperfect knowledge of the system; the formalism ofdensity operator is a convenient way to treat these uncertainties. Hereintentionally limit ourselves to the discussion of wave functions~purestates!.

8Normally, in physics, information~or probabilities! is about something!~Meaning about something which has an independent reality, see fostance Sec. VII of Ref. 28.!

9Proponents of the orthodox interpretation often remark that one is lethe same experimental predictions, independently of the exact positiothis border, so that any conflict with the experiments can be avoided.

10With, of course, the usual proviso: short quotations taken out of tcontext may, sometimes, give a superficial view on the position of thauthors.

11Later, Heisenberg took a more moderate attitude and no longer cpletely rejected the idea of wave functions describing some physicaality.

12One could add ‘‘and of external observers.’’13Maybe not so obvious after all? There is an interpretation of quan

mechanics that precisely rests on the idea of never breaking this chainEverett interpretation, which will be discussed in Sec. VI E.

14The title of Ref. 37 is indeed suggestive of this sort of interpretatimoreover, Wigner writes in this reference that ‘‘it follows~from theWigner friend argument! that the quantum description of objects is influenced by impressions entering my consciousness.’’ At the end ofarticle, he also discusses the influence of nonlinearities which woulda limit on the validity of the Schro¨dinger equation, and be indications olife.

15This is for instance the purpose of theories with a modified nonlinSchrodinger dynamics: providing equations of motion where during msurements all probabilities dynamically go to zero, except one that goe1.

16Born’s mistake, therefore, was to confuse assumptions and conclusi17These words are carefully defined by the authors of the theorem; se

beginning of Sec. III B 3.18The contradiction in question occurs through the Bell theorem~which is

therefore sometimes criticized for the same reason!, which was intro-duced as a continuation of the EPR theorem.

19Here we will use the words ‘‘settings’’ and ‘‘parameters’’ indifferently20We are assuming here that the computers are not quantum compute~if

quantum computers can ever be built, which is another question!.


,o

s-e

of

sates

ann

.e,

ea-n--

ddesewe

n-

toof

irir

-e-

the

;

eut

r-to

s.the

21In Bell’s notation, theA functions depend on the settingsa andb as wellas onl.

22Schrodinger used to remark that, if all students of a group always giveright answer to a question chosen randomly by the professor amongthey all necessarily knew the answer to both questions~and not only theone they actually answer!.

23One could add that the EPR disproval of the notion of incompatiobservables implies that, at least, two different settings are considereone of the measurement apparatuses; this should correspond, in Bview, to two different physical realities~every different couplea,b actu-ally corresponds to a different physical reality!, and not to a single one aassumed in the EPR reasoning.

24If Bohr had known the Bell theorem, he could merely have replied to Ethat their logical system was inconsistent~see Sec. IV A 3!!

25In this reference, Wigner actually reasons explicitly in terms of hiddvariables; he considers domains for these variables, which correspogiven results for several possible choices of the settings. But thesemains also correspond to categories of pairs of particles, which is where, we use the notion of categories.

26In terms of the Mendel parable: an observation of a violation of the Binequalities would imply that something inside both peas~maybe a pair ofDNA molecules?! remains in a coherent quantum superposition, withodecoherence, even if the distance between the peas is large.

27In fact, the reasoning just requires that the pair21,11 is never obtained,and does not require any statement about11,21.

28But, if the product is fixed, each of the individual components still flutuates with a 100% amplitude, between results11 and21.

29The ideal GHZ experiment would therefore involve only measuremeof commuting observables, i.e., products measured directly without msuring each factor separately. In practice, it is probably easier to meaeach factor in the product; if all four products are needed, this necessimplies successive measurements of incompatible observables withferent experimental setups; the price to pay, then, is that loopholes suthe ‘‘biased sample loophole’’~Sec. V A! may be opened again in theinterpretation of the results.

30This can easily be checked from the well-known properties of the Pmatrices; the minus sign for the third column comes from the producthe twoi’s, arising from the relationsxsy5 isz ; on the other hand, in thethird line one getsi 3(2 i )51 because of the change of order of thoperators.

31Another intuitive way to understand why experiments where most pgo undetected are useless for a violation of the inequality is the followIf one associates zero with the absence of result, the occurrence of mzeros in the results will bring the correlations rates closer to zero andcombination will never exceed 2.

32A perfect correlation between the detections on each side~in an idealexperiment with parametric generation of photons for instance! wouldprovide another possible scheme for a loophole free experiment—thicourse, implies that two channel detectors with a 100% efficiencyused on both ends of the experiment. In itself, the fact that any clicone side is always correlated with a click at the other, independent ofsettingsa andb, is not sufficient to exclude a setting dependence ofensemble of detected pairs. But, if one assumes locality at this stagea simple reasoning shows that a perfect detection correlation is sufficto ensure the independence: How could a particle on one side ‘‘knothat it belongs to the right subensemble for the other particle to betected, without knowing the other setting? In other words, locality arments may be used, not only for the results of the apparatuses~the func-tionsA andB!, but also in order to specify the ensemble of observed p~the distribution functionr!. Under these conditions, the observation~insome future experiment! of a violation of the Bell inequalities with aperfect detection correlation would be sufficient to exclude local theorand therefore to close the loophole.

33For instance, in the proof that makes uses of a probability densityr(l),if one assumes thata and b become two functionsa(l) and b(l), itmakes no sense to compare the average values for different fixed vof a andb.

34In an all-or-nothing coherent state, all spins are not necessarily up infirst component of the state vector, while they are down in the secowhat matters is that every spin changes component from one compoto the other and reaches an orthogonal state~the quantization axis ofevery spin is not even necessarily the same!.

35We could also have assumed that the photon is focused so that it

697F. Laloe

he

acyst

ntregn

ctir iatrn

su

ulbilstackth

the

o

, w

aso-tutsuae oavles

pipo

avtraterngao

eco

rs

otng

uilsitthepti

ewinen

frat

not

e ofce the, thethey

f acan

ameizinglly

e-e-

cs

s,’’

lem

atod.

,’’

of

of

m

ys.

-

m

er-

ics

he

e-

interact only with one sort of atoms, but is not scattered by the otwithout changing the conclusion of this discussion.

36The formalism of the density operator, or matrix, is elegant and compbut precisely because it is compact it sometimes partially hides the phcal origin of the mathematical terms. The density matrix allows onetreat in the same way classical probabilities, arising from nonfundameuncertainties and imperfect knowledge of a physical system, and puquantum probabilities which are more fundamental and have nothindo with any particular observer. But mathematical analogies shouldobscure conceptual difficulties!

37For filtering a spin state, one obviously needs to use a nondestrumethod for detection after the Stern–Gerlach magnet. One could fostance imagine a laser detection scheme, designed in such a way thatom goes through an excited state, and then emits a photon by retuto the same internal ground state~closed optical pumping cycle—this ispossible for well-chosen atomic transition and laser polarization!.

38Here, we assume that all measurements are ideal; if nonideal meaments are considered, a more elaborate treatment is needed.

39This can be done for instance by successive applications of the postof the wave packet reduction and the evaluation of conditional probaties. Note that we have not restored the norm of any intermediatevector to 1, as opposed to what one usually does with the wave pareduction; this takes care of intermediate probabilities and explainssimplicity of result~36!.

40Let U(t,t0) be the unitary operator associated with the evolution ofstate vector between timet0 and time t1 , in the Schro¨dinger point of

view. If P is any operator, one can obtain its transformP(t) in the

‘‘Heisenberg point of view’’ by the unitary transformation:P(t)5U†(t,t0)PU(t,t0), where U†(t,t0) is the Hermitian conjugate ofU(t,t0); the new operator depends in general on timet, even if this is notthe case for the initial operator.

41Using circular permutation under the trace, one can in fact suppress

of the extreme projectorsPN(n;t2) in formula ~37!, but not the others.42The components of the electromagnetic field are vectors while, here

are dealing with scalar fields; but this is unessential.43In Bohm’s initial work, a Newton law for the particle acceleration w

written in terms of a ‘‘quantum potential.’’ Subsequent versions of Bhmian mechanics discarded the quantum potential in favor of a quanvelocity term directly providing a contribution to the velocity. Both poinof view are nevertheless consistent. An unexpected feature of the qtum velocity term is that it depends only on the gradient of the phasthe wave function, not on its modulus. Therefore, vanishingly small wfunctions may have a finite influence on the position of the particwhich can be seen as a sort of nonlocal effect.

44Another unusual effect takes place for a particle with spin: The sdirection associated with the position of the particle may sometimes staneously flip its direction, without any external coupling~Ref. 144!.

45One sometimes introduces the notion of the ‘‘empty part of the wfunction’’ to characterize the wave packet which does not contain ajectory of the particle, for instance in one arm of a Mach Zehnder inferometer. In the present case, this empty part would deposit somethi~aphoton?! in the cavity that, later, would influence the trajectory ofsecond particle—in other words we would have an indirect influencethe empty part on the second particle.

46For ensemble of systems, the discontinuities are averaged, and one rers continuous equations of evolution for the density operator. Since mof the discussion of Ref. 13 is given in terms of density operatomatrices, and of the appearance of statistical mixtures~decoherence!, onemay get the~incorrect! impression that individual realizations are nconsidered in this work; but this is in fact not the case and ‘‘hittiprocesses’’ are indeed introduced at a fundamental level.

47For these nonexclusive families, relation~42! no longer holds since itwould involve double counting of possibilities.

48For instance, one sometimes invokes the practical impossibility to ban apparatus that would distinguish between a macroscopic superpoand the orthogonal superposition; this would justify the elimination ofcorresponding histories from those that should be used in the descriof reality. Such an argument reintroduces the notion of measuremapparatus and observers in order to select histories, in contradictionthe initial motivations of this point of view—see Rosenfeld’s citationSec. II. Moreover, this again immediately opens the door to Wigner fritype paradoxes, etc.

49Nevertheless, since the Everett interpretation completely suppressesthe beginning any specific notion of measurement, measuring appar


r,

t,i-

oallytoot

ven-the

ing

re-

atei-teete

ne

e

m

n-f

e,

nn-

e--

f

ov-st/

dion

onntith

d

omus,

etc., the usefulness of completing it with the history interpretation isobvious.

50Note that what is envisaged here is communication through the choicthe settings of the measurement apparatuses; this makes sense sinsettings are chosen at will by the experimenters; on the other handresults of the experiments are not controlled, but random, so thatcannot be directly used as signals.

51The ‘‘cloning’’ operation is not to be confused with the preparation oseries of particles into the same known quantum state: This operationbe performed by sending many spin 1/2 half particles through the sStern–Gerlach magnet, or many photons through the same polarfilter. What is theoretically impossible is to perfectly duplicate an initiaunknown~and arbitrary! state.

REFERENCES „indicated in the text by ‘‘Ref.’’ …

a!Electronic mail: [email protected]. Bohr, ‘‘The Solvay meetings and the development of quantum mchanics,’’ Essays 1958–62 on Atomic Physics and Human Knowledg~Vintage, New York, 1966!; Atomic Physics and the Description of Nature ~Cambridge U.P., Cambridge, 1930!.

2J. von Neumann,Mathematical Foundations of Quantum Mechani~Princeton U.P., Princeton, 1955!.

3J. S. Bell, ‘‘On the problem of hidden variables in quantum mechanicRev. Mod. Phys.38, 447–452~1966!; reprinted inQuantum Theory ofMeasurement, edited by J. A. Wheeler and W. H. Zurek~Princeton U.P.,Princeton, 1983!, pp. 396–402.

4D. Bohm and J. Bub, ‘‘A proposed solution of the measurement probin quantum mechanics by a hidden variable theory,’’ Rev. Mod. Phys.38,453–469~1966!.

5D. Bohm and J. Bub, ‘‘A refutation of the proof by Jauch and Piron thhidden variables can be excluded in quantum mechanics,’’ Rev. MPhys.38, 470–475~1966!.

6N. D. Mermin, ‘‘Hidden variables and the two theorems of John BellRev. Mod. Phys.65, 803–815~1993!; in particular see Sec. III.

7J. S. Bell, Speakable and Unspeakable in Quantum Mechanics~Cam-bridge U.P., Cambridge, 1987!; this lovely little book contains the com-plete set of J. Bell’s articles on quantum mechanics.

8A. Shimony, ‘‘Role of the observer in quantum theory,’’ Am. J. Phys.31,755–773~1963!.

9D. Bohm, ‘‘A suggested interpretation of the quantum theory in terms‘hidden’ variables,’’ Phys. Rev.85, 166–179 ~1952!; 85, 180–193~1952;!; see alsoQuantum Theory~Constable, London, 1954!, althoughthis book does not discuss theories with additional variables.

10N. Wiener and A. Siegel, ‘‘A new form for the statistical postulatequantum mechanics,’’ Phys. Rev.91, 1551–1560~1953!; A. Siegel andN. Wiener, ‘‘Theory of measurement in differential space quantutheory,’’ ibid. 101, 429–432~1956!.

11P. Pearle, ‘‘Reduction of the state vector by a non-linear Schro¨dingerequation,’’ Phys. Rev. D13, 857–868~1976!.

12P. Pearle, ‘‘Toward explaining why events occur,’’ Int. J. Theor. Ph18, 489–518~1979!.

13G. C. Ghirardi, A. Rimini, and T. Weber, ‘‘Unified dynamics for microscopic and macroscopic systems,’’ Phys. Rev. D34, 470–491~1986!;‘‘Disentanglement of quantum wave functions,’’36, 3287–3289~1987!.

14B. S. de Witt, ‘‘Quantum mechanics and reality,’’ Phys. Today23, 30–35~September 1970!.

15R. B. Griffiths, ‘‘Consistent histories and the interpretation of quantumechanics,’’ J. Stat. Phys.36, 219–272~1984!.

16S. Goldstein, ‘‘Quantum Theory without observers,’’ Phys. Today51,42–46~March 1998!; 51, 38–41~April 1998!.

17‘‘Quantum mechanics debate,’’ Phys. Today24, 36–44 ~April 1971!;‘‘Still more quantum mechanics,’’24, 11–15~October 1971!.

18B. S. de Witt and R. N. Graham, ‘‘Resource letter IQM-1 on the intpretation of quantum mechanics,’’ Am. J. Phys.39, 724–738~1971!.

19M. Jammer, The Conceptual Development of Quantum Mechan~McGraw–Hill, New York, 1966!; second edition~1989!. The same au-thor has written another interesting book on a similar subject:The Phi-losophy of Quantum Mechanics~Wiley, New York, 1974!.

20O. Darrigol,From c-numbers to q-numbers; the Classical Analogy in tHistory of Quantum Theory~University of California Press, 1992!.

21B. d’Espagnat,Conceptual Foundations of Quantum Mechanics~Ben-jamin, New York, 1971!.

22B. d’Espagnat,Veiled Reality; an Analysis of Present Day Quantum M

698F. Laloe

ar

c

D

in

,’ ’is

ne

ppe

ce

t,

’’

e-

.

de

ity

the

vo

m

e

ks,

,’’

d

bili-

den

tal

ri-

ri-ion

tic

er-

ionium:

–by

in

e-

tt.

ll’s

t.

tt.

in-

ofs,’’

d

syear

s

chanics Concepts~Addison–Wesley, Reading, MA, 1995!; Le Reel Voile,Analyse des Concepts Quantiques~Fayard, Paris, 1994!; Une IncertaineRealite, la Connaissance et la Dureé ~Gauthier-Villars, Paris, 1985!; A laRecherche du Reél ~Gauthier-Villars, Bordas, 1979!.

23A. Pais, ‘‘Einstein and the quantum theory,’’ Rev. Mod. Phys.51, 863–911 ~1979!.

24O. Darrigol, ‘‘Strangeness and soundness in Louis de Broglie’s eworks,’’ Physis30, 303–372~1993!.

25E. A. Cornell and C. E. Wieman, ‘‘The Bose–Einstein condensate,’’ SAm. 278, 26–31~March 1998!.

26W. Heisenberg,The Physical Principles of the Quantum Theory~Univer-sity of Chicago Press, Chicago, 1930!.

27P. A. M. Dirac, The Principles of Quantum Mechanics~Oxford U.P.,Oxford, 1958!.

28H. P. Stapp, ‘‘S-matrix interpretation of quantum theory,’’ Phys. Rev.3, 1303–1320~1971!.

29A. Peres, ‘‘What is a state vector?,’’ Am. J. Phys.52, 644–650~1984!.30B. G. Englert, M. O. Scully, and H. Walther, ‘‘Quantum erasure

double-slit interferometers with which-way detectors,’’ Am. J. Phys.67,325–329~1999!; see the first few lines of Sec. IV.

31C. A. Fuchs and A. Peres, ‘‘Quantum theory needs no ‘interpretationPhys. Today53, 70–71~March 2000!; see also various reactions to thtext in the letters of the September 2000 issue.

32C. Chevalley, ‘‘Niels Bohr’s words and the Atlantis of Kantianismin,’’ iNiels Bohr and Contemporary Philosophy, edited by J. Faye and H. Fols~Kluwer, Dordrecht, 1994!, pp. 33–57.

33J. S. Bell, ‘‘Bertlmann’s socks and the nature of reality,’’ J. Phys. C2,41–62~1981!.

34J. S. Bell,Quantum Mechanics for Cosmologists, edited by C. Isham, R.Penrose, and D. Sciama in Quantum Gravity Vol. 2~Clarendon, Oxford,1981!, pp. 611–637; p. 117 of Ref. 7.

35L. Rosenfeld, ‘‘The measuring process in quantum mechanics,’’ SuProg. Theor. Phys.222, Extra Number ‘‘Commemoration issue for ththirtieth anniversary of the meson theory by Dr. H. Yukawa’’~1965!.

36M. Brune, E. Hagley, J. Dreyer, W. Maıˆtre, A. Maali, C. Wunderlich, J.M. Raimond, and S. Haroche, ‘‘Observing the progressive decoherenthe ‘meter’ in a quantum measurement,’’ Phys. Rev. Lett.77, 4887–4890~1996!.

37E. P. Wigner, ‘‘Remarks on the mind-body question,’’ inThe ScientistSpeculates, edited by I. J. Good~Heinemann, London, 1961!, pp. 284–302; reprinted in E. P. Wigner,Symmetries and Reflections~Indiana U.P.,Bloomington, 1967!, pp. 171–184.

38E. P. Wigner, ‘‘The problem of measurement,’’ Am. J. Phys.31, 6–15~1963!; reprinted inSymmetries and Reflections~Indiana U.P., Blooming-ton, 1967!, pp. 153–170; or again inQuantum Theory of Measuremen,edited by J. A. Wheeler and W. H. Zurek~Princeton U.P., Princeton1983!, pp. 324–341.

39E. Schrodinger, ‘‘Die gegenwa¨rtige situation in der quantenmechanik,Naturwissenschaften23, 807–812~1935!; 23, 823–828~1935!; 23, 844–849 ~1935!. English translation: ‘‘The present situation in quantum mchanics,’’ Proc. Am. Philos. Soc.124, 323–338~1980!; or pp. 152–167in Quantum Theory of Measurement, edited by J. A. Wheeler and W. HZurek ~Princeton U.P., Princeton, 1983!.

40E. Schrodinger, Proc. Cambridge Philos. Soc.31, 555 ~1935!; 32, 446~1936!.

41A. Einstein, B. Podolsky, and N. Rosen, ‘‘Can quantum-mechanicalscription of physical reality be considered complete?,’’ Phys. Rev.47,777–780~1935!; or in Quantum Theory of Measurement, edited by J. A.Wheeler and W. H. Zurek~Princeton U.P., Princeton, 1983!, pp. 138–141.

42M. Born ed.,The Einstein–Born Letters (1916–1955) ~MacMillan, Lon-don, 1971!.

43N. Bohr, ‘‘Quantum mechanics and physical reality,’’ Nature~London!136, 65 ~1935!; ‘‘Can quantum-mechanical description of physical realbe considered complete?,’’ Phys. Rev.48, 696–702~1935!.

44D. Mermin, ‘‘Is the moon there when nobody looks? Reality andquantum theory,’’ Phys. Today38, 38–47~April 1985!.

45D. Home and F. Selleri, ‘‘Bell’s theorem and EPR paradox,’’ Riv. NuoCimento14, 1–95~1991!.

46P. Pearle, ‘‘Alternative to the orthodox interpretation of quantutheory,’’ Am. J. Phys.35, 742–753~1967!.

47J. S. Bell, ‘‘On the Einstein–Podolsky–Rosen paradox,’’ Physics~LongIsland City, N.Y.! 1, 195–200~1964!.


ly

i.

’

l.

of

-

48F. Laloe, ‘‘Les surprenantes pre´dictions de la mećanique quantique,’’Recherche182, 1358–1367~November 1986!.

49F. Laloe, ‘‘Cadre general de la mećanique quantique; les objections dEinstein, Podolsky et Rosen,’’ J. Phys. Colloq.C-2, 1–40 ~1981!. Seealso the following articles, especially that by J. Bell on Bertlmann socwhich is a classics!

50D. Bohm,Quantum Theory~Prentice–Hall, Englewood Cliffs, NJ, 1951!.51P. Eberhard, ‘‘Bell’s theorem and the different concepts of locality

Nuovo Cimento B46, 392–419~1978!.52J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, ‘‘Propose

experiment to test local hidden-variables theories,’’ Phys. Rev. Lett.23,880–884~1969!.

53A. Peres, ‘‘Unperformed experiments have no results,’’ Am. J. Phys.46,745–747~1978!.

54J. A. Wheeler, ‘‘Niels Bohr in Today’s Words’’ inQuantum Theory andMeasurement, edited by J. A. Wheeler and W. H. Zurek~Princeton U.P.,Princeton, 1983!.

55E. P. Wigner, ‘‘On hidden variables and quantum mechanical probaties,’’ Am. J. Phys.38, 1005–1009~1970!.

56S. J. Freedman and J. F. Clauser, ‘‘Experimental test of local hidvariable theories,’’ Phys. Rev. Lett.28, 938–941~1972!; S. J. Freedman,thesis, University of California, Berkeley; J. F. Clauser, ‘‘Experimeninvestigations of a polarization correlation anomaly,’’ Phys. Rev. Lett.36,1223 ~1976!.

57E. S. Fry and R. C. Thompson, ‘‘Experimental test of local hidden vaable theories,’’ Phys. Rev. Lett.37, 465–468~1976!.

58M. Lamehi-Rachti and W. Mittig, ‘‘Quantum mechanics and hidden vaables: A test of Bell’s inequality by the measurement of spin correlatin low energy proton-proton scattering,’’ Phys. Rev. D14, 2543–2555~1976!.

59A. Aspect, P. Grangier, and G. Roger, ‘‘Experimental tests of realislocal theories via Bell’s theorem,’’ Phys. Rev. Lett.47, 460–463~1981!;‘‘Experimental realization of Einstein–Podolsky–Bohm gedankenexpiment: A new violation of Bell’s inequalities,’’49, 91–94~1982!.

60W. Perrie, A. J. Duncan, H. J. Beyer, and H. Kleinpoppen, ‘‘Polarizatcorrelations of the two photons emitted by metastable atomic deuterA test of Bell’s inequality,’’ Phys. Rev. Lett.54, 1790–1793~1985!.

61T. E. Kiess, Y. E. Shih, A. V. Sergienko, and C. O. Alley, ‘‘EinsteinPodolsky–Rosen–Bohm experiments using light quanta producedtype-II parametric conversion,’’ Phys. Rev. Lett.71, 3893–3897~1993!.

62J. S. Bell, contribution to ‘‘Foundations of quantum mechanics,’’Pro-ceedings of the International School of Physics Enrico Fermi~Academic,New York, 1971!, course II, p. 171.

63J. P. Jarrett, ‘‘On the Physical Significance of the Locality Conditionsthe Bell Arguments,’’ Noüs 18, 569–589~1984!.

64L. E. Ballentine and J. P. Jarrett, ‘‘Bell’s theorem: Does quantum mchanics contradict relativity?,’’ Am. J. Phys.55, 696–701~1987!.

65A. Shimony, ‘‘Events and processes in the quantum world,’’Quantumconcepts in Space and Time, edited by R. Penrose and C. J. Isham~Ox-ford U.P., Oxford, 1986!, pp. 182–203.

66J. D. Franson, ‘‘Bell inequality for position and time,’’ Phys. Rev. Le62, 2205–2208~1989!.

67J. Brendel, E. Mohler, and W. Martienssen, ‘‘Experimental test of Beinequality for energy and time,’’ Eur. Phys. Lett.20, 575–580~1993!.

68N. Gisin, ‘‘Bell’s inequality holds for all non-product states,’’ Phys. LetA 154, 201–202~1991!.

69S. Popescu and D. Rohrlich, ‘‘Generic quantum non locality,’’ Phys. LeA 166, 293–297~1992!.

70L. Hardy, ‘‘Quantum mechanics, local realistic theories, and Lorentzvariant realistic theories,’’ Phys. Rev. Lett.68, 2981–2984~1992!.

71N. D. Mermin, ‘‘What’s wrong with this temptation?,’’ Phys. Today47,9–11~June 1994!; ‘‘Quantum mysteries refined,’’ Am. J. Phys.62, 880–887 ~1994!.

72D. Boschi, S. Branca, F. De Martini, and L. Hardy, ‘‘Ladder proofnonlocality without inequalities: Theoretical and experimental resultPhys. Rev. Lett.79, 2755–2758~1997!.

73S. Goldstein, ‘‘Nonlocality without inequalities for almost all entanglestates for two particles,’’ Phys. Rev. Lett.72, 1951–1954~1994!.

74D. M. Greenberger, M. A. Horne, and A. Zeilinger,Bell’s Theorem,Quantum Theory, and Conceptions of the Universe, edited by M. Kafatos~Kluwer, Dordrecht, 1989!, pp. 69–72; this reference is not always eato find, but one can also read the following article, published one ylater.

75D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, ‘‘Bell’

699F. Laloe

,’’

anA

ge

naa

x-nte

M.en

o

a

et

97

on

sa

the

in-

er’

7.ne

m

n

a

,

cy

i-e

ot

J.C.

ev.

hy

,’’

ys.

L.and

.

ite

al-

and

ay

u-

ni-

ions,

on

,’’

uter

v.

m-

, andr-

s,. A

eys.

or

ht-

so

in

-.

theorem without inequalities,’’ Am. J. Phys.58, 1131–1143~1990!.76F. Laloe, ‘‘Correlating more than two particles in quantum mechanics

Curr. Sci.68, 1026–1035~1995!.77N. D. Mermin, ‘‘Quantum mysteries revisited,’’ Am. J. Phys.58, 731–

733 ~1990!.78R. Laflamme, E. Knill, W. H. Zurek, P. Catasi, and S. V. S. Mariapp

‘‘NMR GHZ,’’ quant-ph/9709025; Philos. Trans. R. Soc. London, Ser.356, 1941–1948~1998!.

79Seth Lloyd, ‘‘Microscopic analogs of the Greenberger–Horne–Zeilinexperiment,’’ Phys. Rev. A57, R1473–1476~1998!.

80U. Sakaguchi, H. Ozawa, C. Amano, and T. Fokumi, ‘‘Microscopic alogs of the Greenberger–Horne–Zeilinger experiment on an NMR qutum computer,’’ Phys. Rev. A60, 1906–1911~1999!.

81Jian-Wei Pan, Dik Bouwmeester, H. Weinfurter, and A. Zeilinger, ‘‘Eperimental entanglement swapping: Entangling photons that never iacted,’’ Phys. Rev. Lett.80, 3891–3894~1998!.

82Jian-Wei Pan, Dik Bouwmeester, H. Weinfurter, A. Zeilinger, andDaniell, ‘‘Observation of three-photon Greenberger–Horne–Zeilingertanglement,’’ Phys. Rev. Lett.82, 1345–1349~1999!.

83N. D. Mermin, ‘‘Extreme quantum entanglement in a superpositionmacroscopically distinct states,’’ Phys. Rev. Lett.65, 1838–1841~1990!.

84S. Kochen and E. P. Specker, ‘‘The problem of hidden variables in qutum mechanics,’’ J. Math. Mech.17, 59–87~1967!.

85F. Belifante,Survey of Hidden Variables Theories~Pergamon, New York,1973!.

86A. Peres, ‘‘Incompatible results of quantum measurements,’’ Phys. LA 151, 107–108~1990!.

87J. S. Bell, oral presentation to the EGAS conference in Paris, July 1~published in an abridged version in the next reference!.

88J. S. Bell, ‘‘Atomic cascade photons and quantum mechanical nlocality,’’ Comments At. Mol. Phys.9, 121 ~1980!; CERN preprintTH.2053 and TH 2252.

89P. G. Kwiat, P. H. Eberhard, A. M. Steinberg, and R. Chiao, ‘‘Propofor a loophole-free Bell inequality experiment,’’ Phys. Rev. A49, 3209–3220 ~1994!.

90E. S. Fry, T. Walther, and S. Li, ‘‘Proposal for a loophole-free test ofBell inequalities,’’ Phys. Rev. A52, 4381–4395~1995!.

91A. Aspect, J. Dalibard, and G. Roger, ‘‘Experimental tests of Bell’sequalities using time varying analyzers,’’ Phys. Rev. Lett.49, 1804–1807~1982!.

92G. Weihs, T. Jennenwein, C. Simon, H. Weinfurter, and A. Zeiling‘‘Violation of Bell’s inequality under strict Einstein locality conditions,’Phys. Rev. Lett.81, 5039–5043~1998!.

93J. S. Bell, ‘‘Free variables and local causality,’’ pp. 100–104 in Ref.94H. P. Stapp, ‘‘Whiteheadian approach to quantum theory and the ge

alized Bell’s theorem,’’ Found. Phys.9, 1–25 ~1979!; ‘‘Bell’s theoremand the foundations of quantum physics,’’ Am. J. Phys.53, 306–317~1985!.

95H. P. Stapp, ‘‘Nonlocal character of quantum theory,’’ Am. J. Phys.65,300–304 ~1997!; ‘‘Meaning of counterfactual statements in quantuphysics,’’ 66, 924–926~1998!.

96D. Mermin, ‘‘Nonlocal character of quantum theory?,’’ Am. J. Phys.66,920–924~1998!.

97M. Redhead,Incompleteness, Nonlocality and Realism~Clarendon, Ox-ford, 1988!, Chap. 4.

98J. F. Clauser and A. Shimony, ‘‘Bell’s theorem: Experimental tests aimplications,’’ Rep. Prog. Phys.41, 1881–1927~1978!.

99B. d’Espagnat, ‘‘Nonseparability and the tentative descriptions of reity,’’ Phys. Rep.110, 201–264~1984!.

100B. d’Espagnat,Reality and the Physicist~Cambridge U.P., Cambridge1989!.

101R. B. Griffiths, ‘‘Consistent quantum counterfactuals,’’ Phys. Rev. A60,R5–R8~1999!.

102J. J Bollinger, W. M. Itano, and D. J. Wineland, ‘‘Optimal frequenmeasurements with maximally correlated states,’’ Phys. Rev. A54,R4649–4652~1996!.

103E. Hagley, X. Maıˆtre, G. Nogues, C. Wunderlich, M. Brune, J. M. Ramond, and S. Haroche, ‘‘Generation of EPR pairs of atoms,’’ Phys. RA 79, 1–5 ~1997!.

104K. Molmer and A. Sorensen, ‘‘Multiple particle entanglement of htrapped ions,’’ Phys. Rev. Lett.82, 1835–1838~1999!.

105C. A. Sackett, D. Klepinski, B. E. King, C. Langer, V. Meyer, C.Myatt, M. Rowe, O. A. Turchette, W. M. Itano, D. J. Wineland, and


,

r

-n-

r-

-

f

n-

t.

9

-

l

,

r-

d

l-

v.

Monroe, ‘‘Experimental entanglement of four particles,’’ Nature~Lon-don! 404, 256–259~2000!.

106A. Ekert, ‘‘Quantum cryptography based on Bell’s theorem,’’ Phys. RA 67, 661–663~1991!.

107C. H. Bennett, G. Brassard, and N. D. Mermin, ‘‘Quantum cryptograpwithout Bell’s theorem,’’ Phys. Rev. Lett.68, 557–559~1992!

108W. K. Wooters and W. H. Zurek, ‘‘A single quantum cannot be clonedNature~London! 299, 802–803~1982!.

109D. Dieks, ‘‘Communication by EPR devices, ’’ Phys. Lett. A92, 271–272 ~1982!.

110N. Gisin and S. Massar, ‘‘Optimal quantum cloning machines,’’ PhRev. Lett.79, 2153–2156~1997!.

111C. H. Bennett, G. Brassard, C. Cre´peau, R. Jozsa, A. Peres, and W.Wooters, ‘‘Teleporting an unknown quantum state via dual classicalEinstein–Podolsky–Rosen channels,’’ Phys. Rev. Lett.70, 1895–1898~1993!.

112D. Bouwmeister, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and AZeilinger, ‘‘Experimental quantum teleportation,’’ Nature~London! 390,575–579~1997!.

113S. Massar and S. Popescu, ‘‘Optimal extraction of information from finquantum ensembles,’’ Phys. Rev. Lett.74, 1259–1263~1995!.

114S. Popescu, ‘‘Bell’s inequalities versus teleportation: What is nonlocity?,’’ Phys. Rev. Lett.72, 797–800~1994!; quant-ph/9501020.

115T. Sudbury, ‘‘The fastest way from A to B,’’ Nature~London! 390, 551–552 ~1997!.

116G. P. Collins, ‘‘Quantum teleportation channels opened in RomeInnsbruck,’’ Phys. Today51, 18–21~February 1998!.

117C. H. Bennett, ‘‘Quantum information and computation,’’ Phys. Tod48, 24–30~October 1995!.

118D. P. DiVicenzo, ‘‘Quantum computation,’’ Science270, 255–261~Oc-tober 1995!.

119C. H. Bennett and D. P. DiVicenzo, ‘‘Quantum information and comptation,’’ Science404, 247–255~2000!.

120D. Deutsch, ‘‘Quantum theory, the Church–Turing principle and the uversal quantum computer,’’ Proc. R. Soc. London, Ser. A400, 97–117~1985!.

121P. Shor,Proceedings of the 55th Annual Symposium on the Foundatof Computer Science~IEEE Computer Society Press, Los Alamitos, CA1994!, pp. 124–133.

122D. S. Abrams and S. Lloyd, ‘‘Simulation of many-body Fermi systemsa universal quantum computer,’’ Phys. Rev. Lett.79, 2586–2589~1997!.

123J. I Cirac and P. Zoller, ‘‘Quantum computation with cold trapped ionsPhys. Rev. Lett.74, 4091–4094~1995!.

124P. Shor, ‘‘Scheme for reducing decoherence in quantum compmemory,’’ Phys. Rev. A52, 2493–2496~1995!.

125A. M. Steane, ‘‘Error correcting codes in quantum theory,’’ Phys. ReLett. 77, 793–796~1996!.

126J. Preskill, ‘‘Battling decoherence: The fault-tolerant quantum coputer,’’ Phys. Today52, 24–30~June 1999!.

127C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. SmolinW. K. Wooters, ‘‘Purification of noisy entanglement and faithful telepotation via noisy channels,’’ Phys. Rev. Lett.76, 722–725~1996!.

128C. H. Bennett, D. P. DiVicenzo, J. A. Smolin, and W. A. Wooter‘‘Mixed-state entanglement and quantum error correction,’’ Phys. Rev54, 3824–3851~1996!.

129H. J. Briegel, W. Du¨r, J. I. Cirac, and P. Zoller, ‘‘Quantum repeaters: Throle of imperfect local operations in quantum communication,’’ PhRev. Lett.81, 5932–5935~1998!.

130R. B. Griffiths and Chi-Sheng Niu, ‘‘Semiclassical Fourier transform fquantum computation,’’ Phys. Rev. Lett.76, 3228–3231~1996!.

131S. Haroche and J. M. Raimond, ‘‘Quantum computing: Dream or nigmare?,’’ Phys. Today49, 51–52~August 1996!.

132F. London and E. Bauer,La Theorie de l’observation en Me´caniqueQuantique, No. 775 of Actualite´s Scientifiques et Industrielles, Expose´sde Physique Ge´nerale ~Hermann, Paris, 1939!; translated into English as‘‘The theory of observation in quantum mechanics,’’ inQuantum Theoryof Measurement, edited by J. A. Wheeler and W. H. Zurek~PrincetonU.P., Princeton, 1983!, pp. 217–259; see in particular Sec. 11, but alSecs. 13 and 14.

133E. P. Wigner, ‘‘Interpretation of quantum mechanics,’’ lectures given1976 at Princeton University; later published inQuantum Theory of Mea-surement, edited by J. A. Wheeler and W. H. Zurek~Princeton U.P.,Princeton, 1983!, pp. 260–314; see also Wigner’s contribution in ‘‘Foundations of quantum mechanics,’’Proceedings of the Enrico Fermi Int

700F. Laloe

J.

um

s,’

-

t

nd

nt

c

ys

R-

r r

hy

-cs

s,’

cs

n-ur

it

ertti

ys

c-

s-

Shaeen

v-s,’

n-

nal

vis-

to

ive

s,’’s

a-

.

s,’’

a-

in

ys.

il-

es to

he

tent

v.o

ys.

v.

y,’’

Summer School, edited by B. d’Espagnat~Academic, New York, 1971!.134N. D. Mermin, ‘‘What is quantum mechanics trying to tell us?,’’ Am.

Phys.66, 753–767~1998!.135B. Misra and E. C. G. Sudarshan, ‘‘The Zeno’s paradox in quant

theory,’’ J. Math. Phys.18, 756–763~1977!.136L. de Broglie,Tentative D’interpre´tation Causale et Non-line´aire de la

Mecanique Ondulatoire~Gauthier-Villars, Paris, 1956!.137D. Bohm, ‘‘Proof that probability density approachesuCu2 in causal in-

terpretation of quantum theory,’’ Phys. Rev.108, 1070–1076~1957!.138L. E. Ballentine, ‘‘The statistical interpretation of quantum mechanic

Rev. Mod. Phys.42, 358–381~1970!.139E. Madelung, ‘‘Quantentheorie in hydrodynamische Form,’’ Z. Phys.40,

322–326~1927!.140E. Nelson, ‘‘Derivation of the Schro¨dinger equation from Newtonian me

chanics,’’ Phys. Rev.150, 1079–1085~1966!.141R. Werner, ‘‘A generalization of stochastic mechanics and its relation

quantum mechanics,’’ Phys. Rev. D34, 463–469~1986!.142C. Philippidis, C. Dewdney, and B. J. Hiley, ‘‘Quantum interference a

the quantum potential,’’ Nuovo Cimento Soc. Ital. Fis., B52B, 15–23~1979!.

143J. S. Bell, ‘‘De Broglie–Bohm, delayed choice, double slit experimeand density matrix,’’ Int. J. Quantum Chem., Symp.14, 155–159~1980!.

144B. G. Englert, M. O. Scully, G. Su¨ssmann, and H. Walther, ‘‘SurrealistiBohm trajectories,’’ Z. Naturforsch., A: Phys. Sci.47a, 1175–1186~1992!.

145J. S. Bell, Sec. 4 of Ref. 34 or p. 128 of Ref. 7.146R. B. Griffiths, ‘‘Bohmian mechanics and consistent histories,’’ Ph

Lett. A 261, 227–234~1999!.147N. Gisin, ‘‘Quantum measurements and stochastic processes,’’ Phys.

Lett. 52, 1657–1660~1984!; ‘‘Stochastic quantum dynamics and relativity,’’ Helv. Phys. Acta62, 363–371~1989!.

148L. Diosi, ‘‘Quantum stochastic processes as models for state vectoduction,’’ J. Phys. A21, 2885–2898~1988!.

149P. Pearle, ‘‘On the time it takes a state vector to reduce,’’ J. Stat. P41, 719–727~1985!.

150A. Barchielli, L. Lanz, and G. M. Prosperi, ‘‘A model for the macroscopic description and continual observations in quantum mechaniNuovo Cimento Soc. Ital. Fis., B42B, 79–121~1982!.

151A. Barchielli, ‘‘Continual measurements for quantum open systemNuovo Cimento Soc. Ital. Fis., B74B, 113–138~1983!; ‘‘Measurementtheory and stochastic differential equations in quantum mechaniPhys. Rev. A34, 1642–1648~1986!.

152F. Benatti, G. C. Ghirardi, A. Rimini, and T. Weber, ‘‘Quantum mechaics with spontaneous localization and the quantum theory of measment,’’ Nuovo Cimento Soc. Ital. Fis., B100B, 27–41~1987!.

153P. Pearle, ‘‘Combining stochastic dynamical state-vector reduction wspontaneous localization,’’ Phys. Rev. A39, 2277–2289~1989!.

154G. C. Ghirardi, P. Pearle, and A. Rimini, ‘‘Markov processes in Hilbspace and continuous spontaneous localization of systems of idenparticles,’’ Phys. Rev. A42, 78–89~1990!.

155L. Diosi, ‘‘Continuous quantum measurement and Ito formalism,’’ PhLett. A 129, 419–423~1988!.

156L. Diosi, ‘‘Models for universal reduction of macroscopic quantum flutuations,’’ Phys. Rev. A40, 1165–1174~1989!.

157G. Ghirardi, R. Grassi, and A. Rimini, ‘‘Continuous-spontaneoureduction model involving gravity,’’ Phys. Rev. A42, 1057–1064~1990!.

158Experimental Metaphysics, Quantum Mechanical Studies for Abnermony, Festschrift Vols. 1 and 2, edited by Robert S. Cohen, MichHorne, and John J. Stachel, Boston Studies in the Philosophy of SciVols. 193 and 194~Kluwer Academic, Dordrecht, 1997!.

159F. Benatti, G. C. Ghirardi, A. Rimini, and T. Weber, ‘‘Operations involing momentum variables in non-Hamiltonian evolution equation


’

o

,

.

ev.

e-

s.

,’’

’

,’’

e-

h

cal

.

i-lce

’

Nuovo Cimento Soc. Ital. Fis., B101B, 333–355~1988!.160S. Weinberg, ‘‘Precision tests of quantum mechanics,’’ PRL62, 485–488

~1989!.161K. Wodkiewicz and M. O. Scully, ‘‘Weinberg’s nonlinear wave mecha

ics,’’ Phys. Rev. A42, 5111–5116~1990!.162N. Gisin, ‘‘Weinberg’s non-linear quantum mechanics and supralumi

communications,’’ Phys. Lett. A143, 1–2 ~1990!.163G. C. Ghirardi, ‘‘Quantum superpositions and definite perceptions: En

aging new feasible tests,’’ Phys. Lett. A262, 1–14~1999!.164N. Gisin and I. Percival, ‘‘The quantum-state diffusion model applied

open systems,’’ J. Phys. A25, 5677–5691~1992!; ‘‘Quantum state dif-fusion, localization and quantum dispersion entropy,’’26, 2233–2243~1993!; ‘‘The quantum state diffusion picture of physical processes,’’26,2245–2260~1993!.

165I. Percival,Quantum State Diffusion~Cambridge U.P., Cambridge, 1998!.166M. B. Plenio and P. Knight, ‘‘The quantum-jump approach to dissipat

dynamics in quantum optics,’’ Rev. Mod. Phys.70, 101–141~1998!.167J. G. Cramer, ‘‘The transactional interpretation of quantum mechanic

Rev. Mod. Phys.58, 647–687~1986!; in an appendix, this article containa review of the various interpretations of quantum mechanics.

168R. Omnes, ‘‘Logical reformulation of quantum mechanics. I. Foundtion,’’ J. Stat. Phys.53, 893–932 ~1988!; ‘‘Logical reformulation ofquantum mechanics. II. Interferences and the EPR experiments,’’53,933–955 ~1988!; ‘‘Logical reformulation of quantum mechanics. IIIClassical limit and irreversibility,’’53, 957–975~1988!.

169M. Gell-Mann and J. Hartle, ‘‘Classical equations for quantum systemPhys. Rev. D47, 3345–3382~1993!.

170R. Omnes, The Interpretation of Quantum Mechanics~Princeton U.P.,Princeton, 1994!; Understanding Quantum Mechanics~Princeton U.P.,Princeton, 1999!.

171R. B. Griffiths and R. Omne`s, ‘‘Consistent histories and quantum mesurements,’’ Phys. Today52, 26–31~August 1999!.

172Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, ‘‘Time symmetrythe quantum process of measurement,’’ Phys. Rev. B134, 1410–1416~1964!.

173R. B. Griffiths, ‘‘Consistent histories and quantum reasoning,’’ PhRev. A 54, 2759–2774~1996!.

174R. B. Griffiths, ‘‘Choice of consistent family, and quantum incompatibity,’’ Phys. Rev. A57, 1604–1618~1998!.

175Phys. Today ‘‘Observant readers take the measure of novel approachquantum theory; some get Bohmed,’’52, 11–15 ~February 1999!; 52,89–92~February 1999!.

176W. H. Zurek, ‘‘Preferred states, predictability, classicality and tenvironment-induced decoherence,’’ Prog. Theor. Phys.89, 281–312~1993!.

177R. B. Griffiths, ‘‘Correlations in separated quantum systems: A consishistory analysis of the EPR problem,’’ Am. J. Phys.55, 11–17~1987!.

178F. Dowker and A. Kent, ‘‘Properties of consistent histories,’’ Phys. ReLett. 75, 3038–3041~1995!; ‘‘On the consistent histories approach tquantum mechanics,’’ J. Stat. Phys.82, 1575–1646~1996!.

179A. Kent, ‘‘Quasiclassical dynamics in a closed quantum system,’’ PhRev. A 54, 4670–4675~1996!.

180H. Everett III, ‘‘Relative state formulation of quantum mechanics,’’ ReMod. Phys.29, 454–462~1957!; reprinted inQuantum Theory and Mea-surement, edited by J. A. Wheeler and W. H. Zurek~Princeton U.P.,Princeton, 1983!, pp. 315–323.

181H. D. Zeh, ‘‘On the interpretation of measurement in quantum theorFound. Phys.1, 69–76~1970!.

182H. Hertz, Miscellaneous Papers, translated from first German edition~1895! by D. E. Jones and J. A. Schott~MacMillan, London, 1896!, Vol.1, p. 318.

701F. Laloe

do we really understand quantum mechanics strange correlations

Documents