conditional probability in the quantum theory of measurement

IL NUOVO CIMENTO VOL. 108 B, N. 1 Gennaio 1993

Conditional Probability in the Quantum Theory of Measurement.

G. CASSINELLI(1) and P. J. LAHTI (2)

(1) Dipartimento di Fisica, Universitd di Genova, INFN, Sezione di Genova 16146 Genova, Italia

(2) Department of Physics, University of Turku - 20500 Turku, Finland

(ricevuto il 25 Ottobre 1991; approvato il 6 Aprfle 1992)

Summary. - - The notion of conditional probability is applied to provide alternative probabilistic characterizations of strong-state correlation, strong-value correlation, and ideal measurements. These results will be applied to work out conditional interpretations of probabilities for measurement outcomes in sequences of measurements.

PACS 03.65.Bz - Foundations, theory of measurement, miscellaneous theories.

1 . - I n t r o d u c t i o n .

The notion of conditional probability touches the quantum-theory of measurement in various points. The very idea is that the final states of the object system and the measuring apparatus give rise to conditional probabilities. The fact that in quantum-mechanics the notion of conditional probability is not always additive with respect to a partition of the conditioning event calls for a careful study of that notion in the context of the measurement theory (sect. 3). It turns out that the notion of conditional probability leads to an alternative probabilistic characterization of strong- state correlation measurements (sect. 4), strong-value-correlation measurements (sect. 5) as well as ideal measurements (sect. 5). These results have an immediate application in the theory of sequential measurements, that is, in the context where two (or more) measurements are performed on the object system one after the other in immediate succession. Indeed, interpreting sequential probabilities as probabilities for measurement outcomes for the second measurement with the condition that the first measurement would have led to a result, or even to a particular result, is a typical requirement on conditional probability. The conditional interpretation of the sequential probabilities depends crucially on the condition involved in the first measurement. Some of the most typical cases will be briefly discussed in sect. 6.

It has been argued that the quantum theory of measurement provides a consistency proof for the minimal interpretation of quantum mechanics (see, e.g., ref. [1]). In the same way the interpretation of sequential probabilities as conditional

45

46 G. CASSINELLI and P. J . LAHTI

probabilities, conditionalized with the first measurement, gains its consistency via the quantum theory of measurement supplemented with the theory of conditional probability in quantum mechanics. Similar ideas have been advanced, e.g., in ref. [2, 3].

2 . - P r e l i m i n a r i e s .

2"1. The basic framework. - We shall base our study on the Hilbert-space formulation of quantum mechanics. To fLX the notations and terminology we shall briefly summarize the main items of this theory.

The description of a physical system J" is carried out in a complex, separable Hilbert space ,~f~, with the inner product ( l ). We let J~(.~iJ') denote the set of bounded operators on .~ir, and _d%~:r) + and ~7-(.~:r) its subsets of positive and trace class operators, respectively.

Observables of d' are represented as (and identified with) projection-operator- valued measures E : ~7-~ _~'(.~j.)+ on a measurable sp~ce (t2, ,~7"), the value space of E. Usually, (s .-7-) is the real Borel space (~, .c$(,~)), or a subspace of it, or a C~tes ian product of such spaces.

States of , f are represented as (and identified with) the positive trace-one operators T ~ J-(.~iJ')~. The extremal elements of the convex set ,~ ( .~ r )~ are the one-dimensional projection operators P[~] generated by the unit vectors ~ of .~ir- Such states, as well as the generating unit vectors are called vector states, which all are pure states whenever J" is a proper quantum system.

Any pair (E, T) of an observable E and a state T defines a probability measure ET: , ~ - o [ 0 , 1] through the trace formula ET(X):=tr[TE(X)]. According to the minimal interpretation, the number E r (X) is the probability that a measurement of the observable E on the system J" in the state T leads to a result in the set X.

2"2. Conditional probability. - Let (t2, ~ / z ) be a (classical) probability space, and Y ~ ~ an event for which/z(Y) ~ 0. The conditional probability of X e J with respect to Y is p~ (X I Y):= ~(X N Y)/F(Y). Clearly, p~ (. I Y) is a probability measure on ~J-for which p , ( Y I Y ) = 1. Moreover, it has the property: if X cY, then p,~(XIY)= = F(X)/F(Y). This compatibility property is, in fact, a characteristic property of conditional probability. Indeed, for a given Y~ ~ f~(Y)~ 0, there is exactly one probability measure v on ~ for which v (X)= F(X)/~(Y) for all X r Y, that is, ~ = = P~ (" I Y) [4]. It is to be observed that p~ is also additive with respect to conditioning events. Indeed, if Y = U Y~, Yi N Yj = O for i ~ j , and if [x(Y i ) ~ 0, then p , (" I Y) = = ~ (~(Y~)/[z(Y)) p.~ (. I Yi), that is, p~ (. I Y) is a convex combination (mixture) of the

component measures P~('IYi). This additivity proper ty serves as an alternative characterization of (classical) conditional probability [5]. Clearly, P~ (Yi I Y) = = tz(Y~)/,a(Y) for each i.

According to Gleason's theorem the (generalized) probability measures in quantum mechanics are generated by states T through the formula P ~ t r [ T P ] , P �9 ~(5~.r), P = p2 = p . . Consider a state T, and let R be a projection operator such that t r [TR] ~ O. The conditional probability with respect to R is the (generalized)

CONDITIONAL PROBABILITY IN THE QUANTUM THEORY OF MEASUREMENT 47

probability measure induced by the state

(1) T R := . R T R t r [TR] "

It is the (generalized) probability measure defined by the following property: if P ~< R, then t r [TRP] = tr[TP]/tr[TR][4]. Clearly, tr[TRR] = 1.

Though the conditional probability tr [T R.] is a probability measure, it is not, in general, additive with respect to a partition of the conditioning event. Indeed, if R = = ~ Ri is a decomposition of the condition R into mutually orthogonal conditions

i (projection operators) Ri, and if tr [TRi] ~ 0, then tr[RTRP] = ~ tr[RiTRiP] only if

i P commutes with all Ri with respect to T. The fact that, in general, T R need not be equal to ~( t r [TRi] / t r [TR])T Ri, with TR,=RiTRi / t r [TRi] , implies some further

i care in defining (generalized) conditional probability with decomposed conditions. Indeed, the conditional probability with respect to a partition (Ri) of R (into mutually orthogonal projection operators Ri) is the (generalized) probability measure defined by the (mixed) state

Ri TRi _ ~. t r [TRi ] TR i (2) T(R,) :_- Y "7" t r [TR] "7" t r [TR] "

It is the (generalized) probability measure defined by the property tr[T(R')P] = =tr[TRp] for each projection operator P in any segment [O, Ri][5]. Clearly, tr[T(R')R] = 1 and tr[T(R~)Ri] = tr[TRi]/tr[TR] 'for each i.

2"3. Measurements. - The quantum theory of measurement is a part of the theory of compound systems in quantum mechanics. In its usual formulation, the measurement theory of an observable E of the object system d ~ starts with fixing a measuring apparatus ~ (with a Hilbert space ~ ) , its initial (vector) state

e :~V~, a pointer observable Pal: ~'--~.~(gV~ ), and a measurement coupling U: ~ 2 | :~V~ --, ~ | :~V~ (a unitary operator). The interpretation of ( ~ , P ~ , ~, U) as a measurement of E starts with the assumption that if ~ | ~o- is an initial (vector) state of the object system ~f, then U(~ | ~) is the final (vector) state of the compound object-apparatus system d ~ + ~ r The reduced states

9~2[P[U(~| and ~ [ P [ U ( ~ | are then the final states of d ~ and ~r respectively. As a rule, they are mixed states. Here ~ , 9 ~ denote the partial traces over the apparatus and object system Hilbert spaces 9~-z and :~V~, respectively.

A basic requirement for ( ~ , P~, ~, U) to constitute a measurement of E is the probability reproducibility condition:

(3) (+IE(X)+) = {g(~ | +) IX | P~ (X) U(~ | +)),

for all X e ~ , and for any vector state ~ �9 9~r. Measurements (gVd, Pd , ~, U) which fulfil this condition are known as normal unitary premeasurements of E. For further explanation of the concept of measurement the reader may wish to consult, e.g., ref. [1].

We close this preliminary section with recalling the notion of a reading scale. A reading scale is a countable partition of the value space ~, that is ~ = t3Xi, 2: / �9 95, Xi N Xj = O for i ~ j . Such a reading scale will be denoted as 9~. A reading scale

48 G. CASSINELLI and P. J. LAHTI

determines a discrete, coarse-grained version of the pointer observable P ~,

(4) '~ , �9 P ~(X~). p ~ : i ~ P ~.~.= .

The P:~y-value i refers to the pointer values X~ which, in turn, may refer to the value X~ of the measured observable E. It is with respect to such a reading scale that measurement results are to be recoded.

3 - Conditional final states.

3"1. Conditioning with respect to a pointer value. - Consider a measurement ( , ~ ~, p ~, ~, U) of an observable E. If p is the initial state of d', then U(9 | ~) is the final state of d ' + �9 [. Together with a condition I | ((X), this state defines a (non-normalized) conditional final state o f , ; + , [: I | p ~ (X) P[ U(p | ~)] I | p ~ (X). Up to a normalization factor, this is the final state of , f + ~ { with the condition I ~ p ~ (X), that is, with the condition that the pointer observable P r has the value X. The corresponding final (non-normalized) states of d" and, [ are

(5) c 2 r [ I | 1 7 4 1 7 4 = {,4r[P[U(p|174

and

(6) [2 ( [ I | 1 6 2 1 7 4 1 6 2 =P~(X) Y/? ([P[U(9|162

respectively. The expressions after the equality signs in (5) and (6) are obtained by using che basic properties of the partial trace and the fact that the conditioning event is of the special form I | p r (X). Since partial trace is a trace-preserving mapping, we observe that, due to the probability reproducibility condition, (3) we also have

(9 [E(X) ~} = tr [I | p ~ (X) P[ U(~s | ~)] I | p ~ (X)] =

= tr [c2 r[P[ U(~ | ~b)] I | p ((X)]] = tr [p ~ (X) ,~2 ~[P[ U(9 | ~)]] P ((X)],

for all X �9 ~ and for any ~ �9 -~ir. Let N~ := (~ I E(X) p), and denote

(7) Tj,(X, 9): = N x 2 ,c2 ,.[P[U(p | r I | p ~(X)]

and

(8) T ,,(X, 9) := N x 2P, ~,(X) ( ~ ~[P[U(~s | (P)]] P ((X),

whenever N~ ~ 0. For later use we also define Tr(X, 9 )= O = T t(X, 9) whenever ( 9 [ E ( X ) ; ) = 0 . With these notations Tj,(O, 9) = ~ r [ P [ U ( g Q ~ ) ] ] and To(O, 9) = = ~ , ([P[ U(9 • ~)]]. We call T.r (X, 9) and T ~ (X, 9) the final (X-) component states of d" and , ~', respectively. These states are conditional states, that is, they give rise to conditional probabilities. Indeed, by (1) T~(X, 9) is just the conditional state of , t defined by the final apparatus state T ((O, 9) and the condition P ~(X). In particular, we observe that the conditional probability for p ((X) in the final apparatus state T/(t2, 9) given p ~ ( x ) is, indeed, equal to one (whenever (9[E(X)9} ;~ 0), that is, t r [T ~(X, ~)P/ (X)] = 1. T c(X, 9) is the final state of, [ with the condition P r In this state the pointer observable has the value X in the sense that tr[Tr 9)P c(X)] = 1.


Consider next the object state Tr(X, ~). The conditional interpretation of this state is not as evident as that of T~(X, ~), since the state T~r(X, ~) and the involved condition P~(X) refer to different systems ~r and ~r respectively. In any case, the conditional interpretation of Tr(X, ~) is in full accordance with the classical notion of conditional probability. Indeed, let B �9 ~f'(~j.) be a self-adjoint operator with the spectral m e a s u r e E s. When interpreted as observables of ~r § ~ , the observable B commutes with the pointer observable P~. This implies, in particular, that these observables have the ordinary joint probability in each state of ~r + ~ , in particular in the state U(~ | ~). I t is the unique extension of the mapping Y x Z ~ ( Y x Z):= := (U(~ | ~) lES (y) | p~(z ) U(~ | ~)) [6, 7]. With respect to this (classical) probability measure the conditional probability for the outcome Y x ~ (that is, for EB(y) | I~) in the state U(~ | ~) with the condition t~ x X (that is, with I | P~(X)), X �9 9~, can be given. As recalled in subsect. 2"2, it is

(9) ~(Y x X)

~(t~ x x) (U(~ | ~)IEB(y) | P~(X) U(~ | ~))

(U(~|174174 -tr[Tr ~)EB(y)].

The conditional interpretation of T~r(X, ~) is thus in full accordance with the classical notion of conditional probability. This interpretation of Tr(X, ~) was discussed already in ref.[8]. To summarize: T2(X, ~) is the final state of J ~ with the condition P~(X). Clearly, tr [T~r(X, ~)E(X)] = 1 need not hold now; in this state the measured observable need not have the value corresponding to the measurement result. In general,

(10) T,r(X, ~) ;~ E(X) Tr ~) E(X) tr [Tr ~)E(X)]

3"2. Conditioning with respect to the possible pointer readings. - Consider still a measurement (gV~, P~, ~, U) of E. If ~ is the initial state of ~r, then, for any X �9 ~ , T~r (X, ~) and T~ (X, ~) are the fmal component states of ~f and d , and they admit conditional interpretations as the final ~C/~r states with the condition that the pointer observable P~ has the value X. Fix now a reading scale 9~. For any Xi �9 gig we denote T~(i, ~):= T~r(Xi, ~), T~(i, ~):= T~(Xi, ~), and N~:= N~. The states T2(i, ~) and T~(i, ~) are conditional states. The reading scale ~ defines a partition of the (trivial) condition Pz(D) into (mutually orthogonal) conditions P~ , i : = P~(X~). We determine next the final states of ~r and ~ with respect to the partition (P~, ~) of P~(t~) induced by the reading scale 9~.

According to eq. (2), the conditional state of ~ , defined by the final apparatus state T~(t~, ~) and the condition (reading scale) 9~ is

(11) T~ (9~, ~):= ~ N~T~(i, ~). i

In general, T d ( t ~ , ~) ~ Td(t~, ~), since Td(~, ~) = ~Pd , iTd(t~, ~)P~, j and Td (~, ~) need not commute with each P~, i. ~J

Due to the classical nature of the conditional states T2(X, ~) we, however,

4 - II Nuovo Cimento B

5 0 G. CASSINELLI and e . J . LAHT1

have

(12) T,'(f), ~?) = ~ N ~ T , , ( i , 9) =: T, , ( f2 , ?), i

for any reading scale [2 (and for any initial state ~ of d'). In closing this section, we wish to emphasize that the notion of conditional

probability is in full accordance with the probability reproducibility condition of the quantum theory of measurement. The states Tr(X, ~), T~(X, ~), Tr([2, 9) and T ( ( ~ , 9) admit natural conditional interpretations with respect to a pointer reading X and a reading scale [2, respectively. At the same time the following natural questions arise: under which conditions do the conditional states Tr 9) and T ~ (~(2, 9) coincide, and when, say, the final state Tr(t2, 9) of ~f is the final state of ~f with the condition (E(X~)) induced by ~/?. These questions will be studied in the next two sections.

4. - S t r o n g - s t a t e c o r r e l a t i o n m e a s u r e m e n t s .

Consider a measurement ( .~ ~, p r r U} of an observable E, and fLX a reading scale [2. With respect to this reading scale, if ~ is the initial state of J', then Tj.(i, 9) and T ((i, 9) are the final/-component states of ,J' and. ( . The final apparatus state T ~(t2, 9) can be expressed as

(13) T / ( ~ , 9) = ~ P , , ' , i T , "(~2, ~2)P,,',i �9 ~j

The final state of~ ( with the condition ~2 is, however,

(14) T r 9 )= ~P, t , iTr ?)P (,i. i

The question which we should like to examine next is the following: when do the states (13) and (14) coincide? In other words, under which conditions the final apparatus state equals the final apparatus state with the condition that one of the mutually exclusive pointer readings occurs? We shall show that for a given reading scale c/? and for a given initial state ~ of J" this is the case exactly when the final component states Tj~(i, 9) and T ~(i, 9) are strongly correlated.

Before entering the discussion of the claimed result we remind ourselves that the correlation P(Tr(i, 9), T ~(i, ~), U(~ | ~)) of the component states Tr(i, 9) and T~(i, 9) in the final state U(~ | ~) o f , F + . ~" can be given as the number

(15) (u(~ | ~)1T,,(i, 9) | T r 9) u(~ | ~))

tr [Tj.(t2, ~)T~r(i, 9)] tr[Tr163 9)T ~(i, 9)] A,r A c

where

A~, {tr [Tu,(t2, ~)(Tp(i, ~))2] _ tr [Tr(t2, ~)Tj.(i, ~)]2}1/2,


and

d ~ = {tr [T~t(~, ~)(T~(i, ~))2] _ tr [Td(~9 ' ~)T~(i, ~)]2}~/2.

The above expression for the correlation is due to the fact that the component states are bounded self-adjoint operators. The strong-correlation condition

(16) ~(T~.(i, ~), T+(i, ~), U(~ | ~)) = 1

holds exactly when the probability measures defined by the pairs (T~r(i, 9), T~r(D, 9)) and (Td(i, 9), Td(~, 9)) are linearly dependent[9].

In short, we denote E~ := E(X~) and Pi := P.z,~ = Pd(X~). For each i, let {~ij }j and {q~a }1 be orthonormal bases of Ei(gV~) and Pi (SVd), respectively, so that {~i j }ij and {~k~ }~ are orthonormal bases of the Hilbert spaces ~ r and 5Vd, respectively. With respect to these bases, the state U(~ | ~) can be expressed as

(17) U(~| = ~ , Y a r ~ | it

where

(18)

and

(19) N~ = (U(Ei~ | q~)tI | P[r U(Ei~ | r

(so that ~ N ~ = N~ = (9 I E~ 9)). Here, again, we adopt the convention that ~'a = 0

whenever N~ = 0. From (17) the final states of d" and ~d~ can be computed as

(20) Tr (D, 9) = ~ N~ P[Ya], il

(21) T.(!~, 9) = ~ NaNj.,(ra I]'j.,)l~j.,)(~a I . iljm

Similarly, the/-component states of d" and ~r are:

(22) T~r (i, 9) = N(2 ~ j . [P[I ~ Pi U(~ | ~)]] = N(2 ~ N~ P[]'a ], l

and

(23) T.z (i, 9) = N(2 Pi T~ (1-3, fo) P~ = N~ -2 ~ Na N ~ (]'a I ]%,,) I r I. lm

The final state of ~r with the condition ~ gets the form

(24) ilm

Assume first that, for a given 9, the states (21) and (24) are the same. In the present context this implies that for any l and m , NilNjm(Yil IYjm) = 0 whenever i ~ j . A direct computation then shows that all the moments of the probability measures defined by the pairs (Tj.(i, ~), T~r(~, 9)) and (Tz(i, ~), T~(D, 9)) are the same. But this

52 a . C A S S I N E L L I and P. J . L A H T I

means that the measures are the same, and, a fortiori, linearly dependent. Thus they are strongly correlated.

Assume next that, for a given 9, all the component states Tr(i, 9) and T r 9) are strongly correlated: ,z(Tj.(i, ~), T ~(i, 9), U(? | qs)) = 1 for any i (for which N~ ~ 0). A direct, but laborious calculation of the numbers involved in (15) reveals that then N~Nj,,@~ I~'j,,)= 0 for all 1 and m whenever i ~ j . But then, again, the states (21) and (24) are the same.

Thus we have established the following result:

~" �9 ~, U) be a measurement of an obsewable E. For any Theorem 1. Let( . � 9 {, reading scale ~(/~ and for any initial vector state 9 of the object system, the following three conditions are equivalent:

a) T ((t2, 9) = ~ Pi T ~ (t2, 9) Pi ;

b) ,z(Tj.(i, 9), T ~(i, 9), U(9 | cp)) = i for each i = 1, 2, ... for which N~ ~ O;

c) Tr(i, 9 ) Tr(J, ~) = 0 for i ;~ j.

5. - S trong-va lue corre la t ion and ideal m e a s u r e m e n t s .

Consider a measurement ( ,~ ~, p r q), U) of an observable E. If 9 is the initial state of ,f, then Tj,(X, 9), X � 9 ,~7, is the final X-component state of ~f after the measurement given p ~(X). Moreover, with respect to any reading scale ,(/Z

(25) Tr(t2, 9) = ~ N~ z T,.(i, 9) = Tr (,~/?, 9). i

This additivity condition is just an alternative way to express the fact that a measurement of E induces an instrument, that is, a state-transformation-valued mapping X ~ J ( X ) where

(26) .~(X) P[9] = [2.,.[P[U(9 | ~b)] I | p ((X)], 9 �9 .(~.J'.

Clearly, tr [ F/(X) P[~]] = (9 I E(X) 9) and J ( X ) P[91 = N~ T,.(X, 9). As was already pointed out, the conditional interpretation of Tp(X, 9) does not

imply that tr[Tj,(X, 9)E(X)] = 1. Thus the final X-component state Tj,(X, 9) of ,c need not be the final state of , f with the condition E(X). In a similar way, though Tr(t2, 9) = Tr( ~c~r , 9), the equality Tr(f2, 9) = ~ Ei Tr(~2, 9)El need not hold. In this

i section we shall study the conditions under which the state Tr(t2, 9) admits a conditional interpretation with respect to the partition (El) of E(t2) induced by c2.

There are, at least, two natural ways to tackle the above-posed question. The first, and, perhaps, the most natural approach is to ask when the equality

(27) Tj.(f2, 9) = ~ Ei T,r(f2, p)E~ i

holds for a given reading scale ,~/~ and initial state 9. It appears that this is the case exactly when the values of E and P r associated with f/? are strongly correlated in the final object-apparatus state U(~ | ~).


The other approach, equally natural though more restrictive, is to ask when the equality

(28) T~r(t~, ~) = ~ EiP[~IEi i

holds for a given reading scale ~ . It turns out that this is the case if and only if the measurement is ideal with respect to ~ .

The correlation ~(Ei, Pi, U(~ | ~)) of the values E~ and P~ of E and P~ (with respect to 9~) in the final state U(~ | ~) can now be given as in eq. (15). Since Ei and Pi are projection operators, the strong-correlation condition

(29)

holds exactly when the equality

(30) tr [Tx(t), ~)E~] = tr [T~(t~, ~)P~]

holds true[9]. But now tr[T~((t~, ~ ) P ~ ] = N 2, whereas tr[To.(t~, ~)Ei] = ~ N 2 , " m ~

�9 (~'~. IEi ~',~ ). Thus, in order that (30) be valid, it is necessary that E~ ]'ij = ]qj for each j and Ei ].~. = 0 for all m and n whenever m ~ i. In that case

(31) = v)E , i

so that the final state of ~f admits a conditional interpretation also with respect to the decomposition (Ei).

Assume next that the condition (31) holds true for a given initial state ~ of d~. Then, again, a direct computation shows that the vectors ~./j are eigenvectors of the projection operators Ei. In that case t r [T~(~ ,~)Pi]=tr[T~(D,~)Ei] , which guarantees that the strong-correlation condition (29) is satisfied. To summarize:

Theorem 2. Let (gV~, P~, ~, U) be a measurement of an observable E. For any reading scale ~ and for any initial vector state ~ of the object sytem~ the following three conditions are equivalent:

a) = v)E ; i

b) p(Ei, Pi, U(~ | ~)) = 1 for each i = 1, 2, ... for which N~ ~ 0;

c) Ei T~c(i, ~) = T~(i, q~, ) for each i = 1, 2, .....

To connect this result with the previous one, we observe that if one of the conditions of theorem 2 is satisfied, then any of the conditions of theorem 1 are satisfied as well. In other words, for example, strong-value correlation implies here strong-state correlation.

We consider next the case that the final state T2(D, ~) of the object system not only admits the conditional interpretation with respect to (Ei) but is, in addition, of the form

(32) T~r(t~, ~) = ~ EiP[~]E~


for a given initial vector state 9 of d'. To study this case we need to recall the notion of ideality of a measurement.

Intuitively, a measurement is ideal if it alters the measured system only to the extent which is necessary for the measurement result. In the present, context, the ideality of ( . ~ ~, p r, ~, U) with respect to the reading scale c~ can be expressed a s :

(33) if (~IE~?} = 1, then T,.(i, 9) = P[~],

for all i = 1,2 .... and for all ~ e.~d~.j,[1]. Note that ff (~IE~} = 1, then Tj.(t), ~) = = T,.(i, ~), since all other N~ equal 0 in that case. If <.W ~, p ~, ~, U) satisfies (33), we say that the measurement is ~- ideal . From the results of ref. [1] it follows that if <.W: ~, P l, ~b, U} is an ,c/1 -ideal measurement of E, then, indeed, eq. (32) holds true for any initial state of the object system. On the other hand, if

(34) Tp(~2, ~) = ~ E~ P[?] Ei i

holds for each ~, then any (non-zero) ~,~ equals some ~,~ modulo a phase factor. This then guarantees that (33) hold for ( .~ l, P r ~, U}. We summarize:

Theorem 3. Let <.W: l, P t, r U} be a measurement of an observable E. For any reading scale c2 the following two conditions are equivalent:

a) Tr(t), ~;) = ~, EiP[~]E i for each ~;

b) <.V~. ~, p ~, ~, U} is [2 -ideal.

6. - S e q u e n t i a l m e a s u r e m e n t s .

Let E1 and E2 be any two observables of J ' , and consider any of their measurements. These measurements can be combined to yield a sequential measurement of the two observables, performing them on J ' one after the other in immediate succession, for example, ,frrst E1 and then E2--measurement. The instrument induced by such a sequential E1E2-measurement is the composed instrument [10] Y/2 o g l of the instruments J 1 and .~r2 induced by the measurements of E1 and E2.

In accordance with the minimal interpretation of quantum probabilities, the number tr[,~r2(y)o .~ri (X) P[;;]] is the probability that the sequential E1E2- measurement performed on , f in the state P[9] leads to a result in the product set Y x X. We shall call such probabilities sequential probabilities. Since

(35) tr [ .~2 (Y) o .~rl (X) P[9]] = tr [( J l (X) P[~]) E2 (Y)],

this number can also be interpreted as the probability that a measurement of E2 on ~Y in the (non-normalized) state .~1 (X)P[~] leads to a result in the set Y. According to subsect. 3"1, the (non-normalized) state.~/,(X)P[9] admits a conditional interpretation as the (non-normalized) state of ~J' after the El-measurement with the pointer condition P~(X), given that P[9] is the initial state of d ~. Hence the sequential probability tr [ .~/2 (Y)o .~r (X)PIl l] admits also a conditional interpretation: it is the probability that an Ee-measurement leads to a result in the set Y with the condition


that the preceeding El-measurement on ~f initially in the state P[~] would have led to a pointer value X. In this sense sequential probability always admits a conditional interpretation. The conditioning depends explicitly on the first measurement.

In the theory of sequential measurements one usually aims to interpret sequential probabilities, like tr [( Y~l (X) P[~]) E2 (Y)] in a more direct way as the probability that a measurement of E2 leads to a result in the set Y, given that the preceeding El-measurement performed on ~f in the state P[~] would have led to a result in the set X. In such an interpretation two requirements have to be satisfied: the state transformation, like P[~] ~ J~ (X)P[~], induced by the first measurement must be specified; and, the conditioning related to this transformation is to be specified. The first requirement is always fulfilled when sequential probabilities are formulated within the theory of sequential measurements, as indicated above. The second requirement refers to the conditioning and is thus subject to the novelties of the conditional probability in quantum mechanics. We shall now illustrate these aspects with the two, perhaps most typical, cases: the first measurement is peformed and the trivial result D is recorded; the first measurement is performed and the result is recorded with respect to a given reading scale ~ . We shall distinguish between three different cases.

Case 1. For any ~ e ~ : , J l ( l ) ) P [ ~ ] = ~ 1 ( 0 X~IP[~] = ~ ~I(Xi)P[T] for each ] i

reading scale ~ . The sequential probability tr [( ] 1 (D)P[~])E2 (Y)] can thus always be decomposed as

(36) t r [( -~1 (~) P [~ ] ) E2 (Y)] =

= ~ tr[(Jl(Xi)P[T])E2(Y) ] = ~ N 2 tr [T:(i, ~)E2(Y)], i i

for any reading scale ~ and for any initial state ~ of ~f. Here we have, again, used the notations N 2 = tr[(Y~l(X~)P[~])] = (~]E~(X~)~) and N2Tx(i, ~)= Y~l(Xi)P[~]. The result (36) simply shows that the conditional interpretation of sequential probabilities is always possible, given that the condition refers to the apparatus (pointer values). In that case the conditional (sequential) probabilities are additive (also) with respect to partitionings of the conditions.

Case 2. Though J1 (D) P[~] = E1 (D)(J~ (D) P[~]) E~ (D), the condition Y~l (D) P[~] = = ~ EI(Xi)(J1 (D) P[~]) E1 (X~) need not hold. Thus the sequential probability

i tr [( :V 1 (12) P[~]) E2 (Y)] cannot be given by ~ tr [El (X~)( ~1 (D) P[~]) E1 (X~) E2 (Y)]. In

i other words, the number tr [(:V 1 (12)[~])E2 (Y)] cannot, as a rule, be interpreted as the probability that a measurement of E2 leads to a result in the set Y given that the preceeding El-measurement would have led to one of the mutually exclusive E~-values X~ when the system is initially in the state ~. As it was shown in the preceeding section, the equality

(37) tr [( J1 (l)) P[T]) E2 (Y)] = ~ tr [E 1 (X i )( ~1 (~r~) p[~]) E1 (Xi) E2 (Y)] i

holds true for a given ~ and ~f if and only if all the Ervalues X~ are strongly correlated with the corresponding pointer values.


Case 3. Finally, we remark that the identification of a sequential probability t r [( .r (~)) P[~]) E2 (Y)] as

(38) t r [( J i (t~) P[~]) E2 (Y)] =

= ~ t r [E 1 (X i ) P[~] E1 (Xi) E2 (Y)] = ~ ( ~ I E I (Xi) E2 (Y) E1 (X~) ~}

is even more restrictive than case (37). This identification is valid if and only if the (first) p remeasurement is ideal with respect to the given reading scale.

R E F E R E N C E S

[1] P. BUSCH, P. LAHTI and P. MITTELSTAEDT: The Quantum Theory of Measurement, Lecture Notes in Physics, Vol. 2 (Springer-Verlag, 1991).

[2] J. BUB: The measurements problem of quantum mechanics, in Problems in the Foun- dations of Physics, edited by G. TORALDO DI FRANCIA (North-Holland, 1979), p. 71.

[3] B. C. VAN FRAASSEN: Quantum Mechanics: an Empiricist View (Oxford University Press, Oxford, 1991).

[4] G. CASSINELLI and N. ZANGHI: Nuovo Cimento B, 73, 237 (1983). [5] G. CASSINELLI and N. ZANGHI: Nuovo Cimento B, 79, 141 (1984). [6] V. S. VARADAR~AN: Geometry of Quantum Theory (Springer-Verlag, 1985). [7] K. YLINEN: On a theorem of gudder on joint distrubutions of observables, in Symposium on

the Foundations of Modern Physics 1985, edited by P. LAHTI and P. MITTELSTAEDT (World Scientific, 1985), p. 691.

[8] M. OZAWA: J. Math. Phys., 25, 79 (1984). [9] G. CASSINELLI and P. LAHTI: Nuovo Cimento B, 105, 1223 (1990).

[10] E. B. DAVIES and J. T. LEWIS: Commun. Math. Phys., 17, 239 (1970).

conditional probability in the quantum theory of measurement

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