pure operations and the covering law

Vol. 16 (1979) REPORTS ON MATHEMATICAL PHYSICS No. 1

PURE OPERATIONS AND THE COVERING LAW

WAWRZYNIEC Guz

Institute of Physics, University of Gdarisk, Gdansk, Poland

(Received November 18, 1977; revised April 17, 1978)

The paper is devoted to the study of properties of pure operations. The main results are: Axiomatic characterization of pure operations consisting in a precise formulation of their basic properties and the representation theorem, which shows that pure operations possessing these (axiomatically defined) properties and satisfying the so-called Compatibility Postulate are in fact the usual ones, of special kind, detined by the well-known formula. The other main result of the paper is the justification of the covering law in the logic of propositions, which is shown here to follow from the Compatibility Postulate, the latter being of direct physical signiftcance.

1. Introduction

Among all axioms assumed for the logic of propositions, the most controversial are the complete lattice property, atomisticity, and the covering law, as not being justified from the physical point of view. Although the first two difficulties connected with the complete atomistic lattice structure of the logic seem now to be resolved by a suitable embedding of the logic (see [1], [2], [6], [S]), the covering law is still left without a satisfactory physical justification.

In this paper, which is devoted primarily to the study of the properties of pure operations, we also give a justification for the covering property of the logic by showing that it follows from the so-called compatibility postulate (see Section 4), the latter being of direct physical significance. The compatibility postulate seems to be necessary also from the mathematical view-point, as it connects the structure of the logic of propositions with the structure of the “logic” of pure operations.

The other main topics of the paper are : analysis of the relationship between the covering law and the existence of pure operations (Sections 3 and 4), and next the axiomatic characterization of pure operations which consists in the precise formulation of their basic properties (Section 3) and then proving the representation theorem, which shows that pure operations possessing these (axiomatically required) properties and the compatibility postulate are, in fact, the usual ones defined by the well-known formula (see Sec- tion 4).

11251

126 W. GUZ

2. Preliminaries: postulates, definitions and notation

In this introductory section we briefly recall the basic definitions, concepts and facts of the “quantum logic” approach to quantum axiomatics. The basic objects of this approach

are: the set L of experimentally verifiable propositions (called also questions, events, yes-no measurements) concerning a physical system, called the logic of the system (briefly, a logic), and the set S of states of the system.

The following postulates for Land Sare commonly accepted and seem to be unquestion- able (see, e.g., [lOI):

1. L is an orthomoduiar a-orthoposet, i.e. it is a partially ordered set with the least and the greatest element 0 and 1, respectively, such that

(i) L is orthocomplemented, that is there is a map a --, a’ of L into itself, called the orthocomplementation of L, such that a < b implies b’ < a’, (a’)’ = a, and a~ a’ = 0

(or, equivalently, a v a’ = 1) for all a E L, where the symbols A and v are used to denote the greatest lower bound and the least upper bound in L, respectively;

(ii) L is a-orthocomplete, that is for each sequence a,, a2, . . . of pairwise orthogonal co

propositions there exists in L its least upper bound // ai (we say that two propositions i=l

a, b E L are orthogonal, and write a ib, if a < b’ or, equivalently, b < a’);

(iii) L is orthomodular, i.e. a i b implies b = av c for some c EL, c la. One can easily show (see, e.g., [15]) that the proposition c is uniquely determined by a and b, and is equal to b A a’ = (b’ v a)‘.

CONVENTION: We shall write a+b instead of av b, whenever a-lb, and b-a for bAa’ = (b’+a)‘, when a < b.

2. Any state m E S is, by the definition, a probability measure on L, that is a non-

negative real-valued function on L such that m(1) = 1 and m(,Q ai) = i$ m(aJ for

any sequence a,, a2, . . . of mutually orthogonal propositions. From (iii) it follows readily that a < b implies m(a) < m(b), hence m maps L into the unit interval [0, 11, and (ii) leads immediately to m(0) = 0.

We assume that S is o-convex, i.e. closed under countable convex combinations of its members: if m,, m,, . . . ES and t,, t,, . . . are nonnegative real numbers such that

i$ ti = 1, then th cc

e probability measure m = c timi also belongs to S. The state m i=l

is called a mixture of m, , m2, . . .

The next postulate (see [1], [3], [6], [81) seems to be necessary in order to formulate quantum axiomatics free of the usual troubles that plague the quantum logic approach, that is axioms possessing no physical significance, as e.g., the complete lattice property of L, atomisticity of L, the covering law in L.

PURE OPERATIONS’ AND THE COVERING LAW 127

3. The pure states of the system are in one-t+one correspondence with the atoms of J,. More precisely (see, e.g., [3], [6], [81), there is a bijection s: P -+ A of the set P of pure states onto the set A of all atoms in L such that’

(i) P(G)) = 1, (ii) p(u) = 1 implies a > s(p), p being an arbitrary pure state from P, a E L.

The atomic proposition s(p) is called the support or carrier of p, and denoted by suppp or carrp (see 1171, [14], [I], [51). For a given atom e E A, the pure statep E P with s(p) = e (that is, p = s-l(e)) is often denoted by p = pc.

The following well-known facts (see, e.g., [15D will be used throughout this paper:

LEMMA 2.1. Let a, b E L, and a t, b, i.e. a = a, +c and b = b, +c, where a,, b, and c are pairwise orthogonal. Then there exist a v b, a A b and a A b = c.

L-2.2. Leta,a,,a, ,... EL.Ifat,atforeach i=1,2,...,andtf$at and I==1

ig (aA ai) both exist, then a t, tq ai and

a,(*? ai> = fq @Aail. 3

Finally, it is not difficult to show the following properties of the relation c), called the compatibility relation in L (see, e.g., [15]):

(i) t, is symmetric, that is a t, b implies b t, a,

(ii) a H b implies a f-, b’,

(iii) a _Lb implies a t) b.

3. The covering law and pure operations

Suppose L is an atomic a-orthoposet, that is, we assume that for every non-zero a EL there is an atom e E L such that e < a. We shall say that the covering law holds in L, or that L possesses the covering property, if

(i) for any a E L and any atom e E L there exists av e in L,

(ii) av e > b > a implies either b = a or b = av e.

F%OPOSITION 3.1. Assume the condition (i) to hold in L, L being an atomic orthomodular o-orthoposet. Then the covering property (ii) is equivalent to the following Jauch-Piron condition [9] :

(iii) UEL, a # 0, eeA imply either (eva’)AaEA, when e&u’, or (eva’)Aa = 0, when e < a’.

1 The state m is said to be pure, if it cannot be written as a non-trivial convex combination of two other states.

By an ufom in L we mean a non-zero element e E L such that e 2 a impIies either a = 0 or a = C.

128 W. GUZ

Proof: Assume first the validity of (ii), and then prove (iii). Let us note that (ev a’) A a exists, as eva’ 3 a’, and hence by the orthomodularity

eva’ = (eva’-a’)+a’ = (eva’)Aa+a’. (3.0

From (3.1) one easily finds thatj

.e < a’ 8 (eva’jva = 0,

which proves the second part of the Jauch-Piron condition. Suppose now that e 4 a’; then by (3.1) ( e v a’) A a # 0. We shall show that (e v a’) A a

is an atom. Assume that b d (e v a’) A a, then e v a’ > b + a’ 2 a’, hence, by (ii),

b+a’ = eva’ or b+a’ = a’.

The first equality implies

(eva’)Aa = (b+a’)r\a = (bAa)f(a’Au) = b

by Lemma 2.2, as b < a implies b c+ a (see, e.g., [4]). The second possibility leads to b < a’, which together with b < a gives us b = 0. We thus have shown indeed that (ev a’) A

ha is an atom. Now, in order to show the implication (iii) =E- (ii), assume that the Jauch-Piron condi-

tion is satisfied, and let a EL, a # 0, e E A, e & a. We shall show that av e covers a. Suppose ev a 2 b > a. Note that e 9; a implies that (ev a) A a’ is an atom, by (iii), but (evu)Aa’ B bAa’ = b-u > 0, hence (evu)Aa’ = bra’, and thus eva = (eva-a)+a = (b-a)+a = b. Therefore ev a indeed covers a, and the lemma is proved.

Thus to each non-zero proposition a EL, L being an atomic logic with the covering law holding in it, one can assign the mapping fO: A + A u (0) defined by

an atom, fa(e) := evd-u =

when e $ a’,

O, when e < a’.

Suppose now that the property 3 of Section 2 is also assumed for the pair (L, P). Then the above-defined mapping_& induces, in an obvious way, the transformation (denoted by E,) of the set P of pure states into itself. It is defined on the set of pure states p E P satisfying p(u) > 0 (i.e. the domain of E,, = D(E,) = (p E P: p(u) > 0)) by the formula :

&P := ~-‘(&W+a = ~((S~)~a,)ho.

One can easily show the following properties of E,:

(1) a EL, a # 0, p E D(E,,) imply (E,p)(u) = 1.

(2) a EL, a # 0, p(a) = 1 imply E,p = p.

(3) a, b E L, a, b # 0, a lb, p E D(E,) imply E,p $6 D(EtJ, that is E, (D(E,))nD(EJ = 0, that is (Ebp)(b) = 0.

The proof of statements (l)-(3) is straightforward and we shall omit it. Note also

that (3) follows from (l), as b la implies (E,,p)(b) < (E,p)(a’) = l- (E,p)(a), hence

(E,p)(b) = 0 by (0

PURE OPERATIONS AND THE COVERING LAW 129

Remark: The properties (l)$ (2) were, accepted by Pool [14] as postulates for his pure operations. ‘, ,I 1’ ‘;)

To avoid complications connected with- the domain of E., which varies when a E Z, is changed, one can extend E., in an obvious way, to act on the set Pu (0). More precisely, first we define extended f, as the transformation f.: Au (0) + Au (0) defined by

_C&> := evp’ya’ = (z atom’ and next extend the bijection s: P + A to the bijection (denoted by the same letter s) S: Pu (0) + Au (0) by setting s(0) = 0. (Note that on the left-hand side zero denotes the function L + R1, which is identically equal zero; on the right-hand side zero denotes the least element in L). The extended E,, is now defined as the unique transformation from Pu (0) to Pu (0}, which makes the following diagram commutative;

Au (0) L Au (0)

p$;; __ pv’;;, E,’

that is, for the extended E,, one has

or, explicitly, E,, = s-Ifas,

E7P = S-l((+)v+a) = P(S(,,,,,),.

for every p E Pu (0).

For the extended E,, which maps Pu (0) into itself, one can readily reformulate the properties (l)-(3) as follows :

(1’) a EL, p(u) > 0 imply (&p)(a) = 1,

(2’) a E L, p(u) = 1 imply E,,p = p,

(3’) a, b EL, a, b # 0, a Ib imply E,E,, = 0,

where by p we denote an element of the set Pu (0). Note also that

(3”) a E L, a # 0, p(u) = 0, p E Pu (0) imply E,p = 0.

In fact, p(a) = 0 leads to s(p) < a’, hence E,p = p(SCp)vaSjha = p0 = 0. Also the follow-

ing statements are almost obvious, and we shall state them without proofs:

PROPOSITION 3.2. Let (E,} be an arbitrary famiIy of transformations qf the set Pu (0)

into itself indexed by non-zero propositions from L which satisfy the conditions (l’), (27, (3”). Then the properties (1’) and (3”) imply (3’), and the collection of all conditions (l’), (27, (3”) impIies the following property of E,:

(2”) a, b e L, a # 0, a < ,b imply Eb E, = E,.

130 ,, W. GIJZ

PROPOSITION 3.3. Let (E,,} be a family of mappings from Pv (0) to ,Pu (0) indexed

by non-zero elements of L. Then the set of properties (l’), (27, (3”) is equivalent to the

collection of properties (l’), (2”), (3’), (4), (5), where

(4) E.(O) = 0 for every non-zero a E L,

(5) a E L, p(a) = l,p~Pimply%p=pforsomeb<a.

The list of conditions (13, (2”), (3’), (4), (5) can be rewritten in a more elegant fashion if we extend the correspondence a + E,, onto the whole logic L by putting

E *= 0 0. ,

where O(p) = 0 for all p E Pu {O}.2

The equivalent set of conditions is then

(l*) a o L ~(4 > 0 => b%p)(a) = 1,

(2*) a,beL, a< b*E,E,,= E,,

(3*) a,bEL, aJ_b=sE,Et, = 0,

(4*) E. = 0,

(5*) a e L, p(a) = 1 * Et,p = p for some b 6 a,

where p denotes an arbitrary ekment of Pu (0).

Note that the conditions (2*) and (4*) above imply immediately the property

E.(O) = 0 for all non-zero a E L.

Indeed, by using (4*) and (2*) we find

E,(O) = E,,(E,p) = E,p = 0.

DEFINITION Any family {E,} of mappings from Pu (0) to Pu (01, indexed by non- zero propositions from L and satisfying the conditions (1’), (2”), (3’), (4), will be called the family of weak pure operations associated with the logic L. It will be called the family of pure operations, if it satisfies, additionally, the condition (5). Thus (see Proposition 3.3) the pure operation families are characterized by the validity of (l’), (2’) and (3”).

The next step in simplifying the description of pure operations one can make by ex-

tending each E, onto the set p of unnormalized pure states (which are of the form sp, p being a pure state, and s-a positive real number) completed by adjoining to it the function 0. That is

P= R:* P = (sp: SER:,~EP),

where R: stands for the set of all nonegative real numbers. Note that any nonzero unnormalized pure state f may be uniquely written as f = sp,

where s is a positive real number and p is a pure state, as then s = f(1) and p = f/f(l).

2 This is in accordance with the fact that for; the usual E,, defined on page 129 one has El = Z = the identity mapping of Pu (0) into itself. Therefore, the: correspondence a + E. is, in some: sense, a representation of the logic L into the set of transformations from Pu{O} to Pu(0).

PURE OPERATIONS AND ‘fT-IFl COVERING LAW 131

The unique pure state p; determined .by f will. be denoted by pf. Thus one can ‘write

f = f(l)Pf:

Now let (E,) be a family of transformations of the set Pu (0) into itself, indexed by. non-zero elements! of .,5,, such that E,p # 0, whenever p(a) # 0. Using Ea one: can

define the following map from p to p:

f(a)E,pf, when f # 0,

0, when f = 0. :

(3.2)

Notice the following properties of P,:

(9 (P,f)(l) = f(a), whenever f(a) # 0; l

(ii) P, is positivelyhomogeneous, that is

P&f) = sP,f for all’f eland s > 0.

Indeed, (P,,f )(I) = f(a)(&~J(l) = f( a ), as E,pf is a pure state, since f(a) # 0 leads

to p,(a) # 0, hence E,pr # 0 by the assumption. Further, using the definition (3.2) one finds for s> 0 andj’Z.0

PaWI = P&f(l)p/) = (sf(l)p,)@)E,pf = sf(a)Gp~ = sP=f;

if either s = 0 or f = 0, & have

thus (ii) is proved. P&f) = P,O = 0 = S’ Paf,

Assume now the family. (E,) to satisfy the stronger condition (1’). Then for (P,] the following holds:

(l)aEL,fEF,f(a)>O - (Paf)(a) = f(o) = (P0f)U).

In fact, it then remains to be shown that (POf)(a) = f(u), which is obvious by (I’):

f(a) > 0 *f # 0 * (P,f)(a) =f(a)(E,p/)(a) =f(4,

as p&z) > 0 implies (&~~)(a) = 1 by (1’). .Notice next the following property of’the family {P,,}, when (E,,) satisfies (I’):

(Ia)aEL,fEF,f(a)>O’ *, Paf = f(ti)p, where p is a pure state satisfying p(a). = 1.

Indeed, by (i) we have,

(Pam) =f(a) > 09 i

hence Paf # 0, and therefore

Pelf = (Pif)(QPP,j = f(a)pFg by (i), where

pda) 7 (Pnfl(P,f )il))(4 =’ 1 by (0.

Note that the property (Ia) can’be trivially extended to’the case, where f(a) = 0, f E p.

Summarizing, one can write the following:

132 W.GIJZ ~ ,, ,

(Ib) For any f & P and a E L, a # 0, we can write P,f ,in the form

P.f = f(a)p,

where p is a pure state such that p(a) = 1. Consider now the condition (2”) for {E,} ({E,} satisfying (l’), of course); we then

easily establish for {Pa> the properties:

(II) a EL, a # 0, a < b j PbP,, = P,,,

(III) a, bEL, a,b # 0, aJ_b=> P,,P, = 0.

Moreover, instead of (III) we now have the stronger property:

(III’) aEL, a#O,fGP,f(a)=O=sP,f=O,

which follows directly from the de&rition of P,.

Note that property (III’) is derived without any use of the conditions (3’), (4), since it follows simply from the definition of P,. Therefore, we have established the following fact:

PROPOSITION 3.4. Let thefamily {I?,,} ( w h ere a EL, a # 0) of mappings from Pu (0)

to Pv (0) satisfy the conditions (1’) and (2”). Then the family P,,: P-t z a # 0, de$ned

by the formula (3.2) satisfies:

0 a E L, f E p7 f(a) > 0 = (P,f)(a) = f(a) = (P,f)(l),

(II) aEL, a # 0, a < b*P,P,, = P,,,

(III’) a E L, a # 0, f E p7 f(a) = 0 * P.f = 0,

(PH) For every non-zero a E L the map P, is positively-homogeneous.

Let us note that the properties (I) and (III’) can be rewritten as the following single condition:

(C) (P,f)(a) = f(a) = (P,_f)(l), f E P, a f 0,

or, equivalently, as the condition

(C) P.f = f(a)p for some pure state p satisfying p(a) = l(f E P, a # 0).

Thus, both (I) and (III’) may be expressed equivalently as follows : P. f is proportional to a pure state p such that p(a) = 1, with f(a) as the proportionality coefficient.

The equivalence of (I)+(rII’) and (C) is obvious. To show the equivalence of (C) and (C’), one needs to prove that (C’) implies (C), as the converse implication we have established earlier, and it is also obvious.

Thus, finally, the essential properties of {PO} can be written as follows:

(I*) a E L, a # 0, f E P, .f # 0 imply P,f = f(a)p for some pure state p such that

p(a) = L3

(II*) a, b EL, a # 0, a < b imply PbP,, = P,,

(III*) P,(sf) = sP,f for all a EL, a # 0, .f~ E s E R:.

’ For f= 0 the property (I*) holds trivialiy by (En*).

PURE OPERATIONS AND Tf-IE COVERING LAW 133

DEFINITION. Any family (E,) of mappings from Pu (0) to Pu @I}, indexed by non- zero elements of L and satisfying the conditions (1’) and (2”), will be called the family

of generalized pure operations associated with the logic L (briefly, the family of g.p.0.).

It will be called the famiZy of normal g.p.o., if it satisfies, additionally, the condition (3”). Note that (1’) and (3”) imply the following property of {E,):

p(a) = 0 if and only if E,p = 0,

where pEPu{O), aEL, a # 0. It would now be appropriate to show schematically the relationships between the

different families of pure operations. This is shown in the following diagram:

pure operations weak

I’), (2’9, (3% (4), (5) + pure operations

0 (I’), (21, (3”) (l’), (2’7, (3% (4)

i i normal generalized generalized

pure operations + pure operations

(1% (2”), (3”) (l’), (2”)

Note that when the property 3 of Section 2 is assumed for the proposition-state system (L, S), then the following holds for any family {E,) of generalized pure operations associated with L:

E,,,,p = p for each pure state p E P. (3.3)

In fact, using (C’) we find for the induced family {Pa) that

PWP = p(sO)q = 4, (3.4)

as p@(p)) = 1 ( see 3, Section 2), q being a pure state with q(s(p)) = 1, hence (see 3, Section 2) s(q) < s(p). The latter inequality implies s(q) = s(p), as s(q) and s(p) are atoms, hence q = p, since s is bijective (see 3, Section 2). Thus P,(,,p = p. But, by the definition

of PSQ), P,WP = p(s(P))E,<,,p = &,,P, (3.5)

and therefore we have shown that E,(,,p = p, as claimed. Applying now Proposition 3.3 one finds as a corollary

PROPOSITION 3.5. For any proposition-state system (L, S) satisfying the conditions

1, 2 and 3 of Section 2, each family of weak pure operations associated with L consists,

in fact, of pure operations.

PROP~STION 3.6. For each family {P,,} of mappings from p to p indexed by non-zero

propositions from L and shtisfying the conditions (I), (II), (III’) and (PH), the famzi’y of

mappings E.: Pu (0) + Pu (0) defined by

E,P = P=plp(a) , when p(u) > 0, o

, when p(a) = 0, p B Pu (0) ,

134 W. GUZ

possesses the properties (l’), (2”), (3”), and, of course,

(3.6)

for all f E i? Thus, (E,) is the family of normal g.p.0. associated to L.

Proof: Ad. (1’): Assume p(a) > 0 (p E Pu (01, a E L, a # 0) ; then

(E&a) = (P,p)(~)/p(a) = 1 by 0.

Ad (2”): Assume a < b (a, b E L, a # 0), and let p E Pu (0).

Case (a): p(a) > 0. Then E,E.p = Pt,(E,p), as E,p = P,p/p(u) implies (E,p)(a)

= 1 by (I), hence by (PH) and (II)

Eb&P = pb(&P/P(a)) = pbpap/p(a) = POP/P(a) = &P.

Case (b): p(a) = 0. Then E,p = 0 by the definition, hence

EbE,p = &,O = 0 = E,p.

Ad (3”): The property (3”) is guaranteed directly by the definition of E,,.

Ad (3.6): If f(a) = 0 (f~ Y’, a EL, a # 0), then P,,f = 0 by (III’), and (3.6) is, of course, trivially valid. Suppose S(a> # 0. Then

PIIf = P0(f(l)P,) = f(l)P.Pz

by (PH), hence, by using the definition of E,, one gets

Paf = f(l)p/(a)E,p/,

as p,(a) = f(a)lf(l) # 0, that is

Paf = f(a) E,P~,

as claimed. The proof of the statement is thus complete.

Propositions 3.4 and 3.6 lead immediately to the following corollary:

COROLLARY 3.7. Any family (E,} f o normal g.p.0. associated to the logic L uniquely

determines, via the formula (3.6), the family {Pa} of mappings from P into itself (a E L,

a # 0) satisfying the properties (I), (II), (III’) and (PH), and vice versa.

Also the following statement is obvious, and we shall state it without the proof:

COROLLARY 3.8. Each family (E,) ofpure operations uniquely determines, by the formula

(3.6), the family (P,,} of mappings from P into itserf (a E L, a # 0) satisfying the condi-

tions (I), (II’), (III’) and (PH), and conversely, where the condition (II’) (stronger than

(II)) is:

(II’) aEL, a#O,f#O,f(a)=f(l)*P,f=f.


Now suppose the family (E,) to be defined as follows:

PsTPlP(4 >

E,p:= ”

I,

ivhen p(u) > 0, (q being a fixed pure state such that E*q = q for all

b 2 a), when p(u) = 0, p P 0,

0, w$enp = 0, pEPv(O}.

Then the conditions (1’) and (2”) are, obviously satisfied, however (3”) is no longer valid. Moreover, even the weaker property (3’) does not hold, so {E,) is an example of a family of generalized pure operations, whose members are not weak pure operations. In fact, the proofs of (1’) and (2”), case (a), given in Proposition 3.6 apply now without any change. To prove (2”), case (b), let us assumep(u) = 0 (p E Pu(O}, a E L, a # 0). One can assume without any loss of generality that p # 0 (as for p = 0 one has EaE.p = 0 = E,p by the definition). Then by the definition E,p = q, q being defined above, hence EbE,p = Et,q = q = E,p, whenever b 2 a, as desired.

As for p E P satisfying p(u) = 0 (a E L, a # 0) one has

E,,p = q= a pure state,

we obtained an explicit example of generalized pure operation family not satisfying the condition (3”). .To prove that also the condition (3’) is not satisfied here, assume a Ib

(a, b E: L, a, b # 0) and choose a pure state p with p(u) > 0. Then

&P)@ = V’~P)(~/P@) = 1 ’ by (I), hence

(&p)(b) = 0 9

as b < a’, hence, by the definition,

Ei@,P) = 4 f 0,

which shows that EbE, # 0. Furthermore, let us notice that the equality (3.6) is also no longer valid; it now hold:

for p satisfying p(u) > 0 only. Note finally that the pure state q, with the properties defined above, may exist. It can

easily be verified that such a state exists when E, is defined by the conventional formula

EaP = P(S(p)“d)-d = a pure

(

state, when p(u) > 0, 0, when p(u) = 0.

Indeed, it is then sufficient to put

q = pe = s-‘(e),

where e is an arbitrary (but fixed) atom < a. Then for any b 2 a one finds

as desired. &q = &P, = P(c+b+b’ = PO = 4,

136 W.GUZ

4. Compatibility postulate and its conseq~~ences

We will assume, in accordance with the results obtained in the preceding section, that to every non-zero proposition u E L there corresponds a mapping P, from p to E 3 being the set of all unnormalized pure states completed by adjoining to it the function identically equal zero, such that the family of all P,,‘s possesses the properties (I*)-(III*), or, equivalently, the properties (I), (II), (III’) and (PH). The members of the family (P.} will be called extended pure operations (briefly, e.p.0.).

To make the structure of the logic of propositions agree more precisely with that of the “logic” of extended pure operations we assume the following postulate (see e.g. [141), which we shall call the Compatibility Postulate (briefly, CP):

COMPATIBILITY POSTULATE: a u b, where a, b E L, implies P, P,, = Pb P,, .

Remark: It would be more natural, of course, to assume the property

a ti b =z- E,Eb = E,E,,

but this implies, clearly, the postulate above.

PROPOSITION 4.1. Let {P,,} be an arbitrary family of extended pure operations. Then the Compatibility Postulate is equivalent to the following condition imposed on {P,,>:

(IV*) a, b EL, a # 0, a < b =, P,Pb= BP,.

Moreover, assuming the Compatibility Postulate wejind that a t* b implies P. Pb = Pb P,, = P ahb.

Proof: The implication CP =+ (IV*) is straightforward, since a < b leads to a H b (see, e.g., [4]). To prove the inverse implication assume (IV*) and let a, b E L, a t-) b,

f E E We shall then show that P,,Pbf = PbP,f = P.,,bjI One can assume without any loss of generality (after excluding the trivial case f = 0) that f # 0. Then by (I*) and (III*) one gets

M’b_f = p, (f@)p) = f(b)P,p = f(b)p(a)q, (4.1)

where p, q are pure states such that

p(b) = q(a) = 1. (4.2)

The assumption a f-, b means (see Section 2) that a = a, +c and b = bl +c, where al, b,, c are pairwise orthogonal. Moreover (see, e.g., [15]) c = ah b. From (4.1) we find

P,IP,Pbf =f(b)p(a)Ptilq. (4.3)

But the inequality a, < a implies, by (IV*) and (II*),

P,,P, = P,P,, = PO,, hence

P,,lP,,P, = P,,IPb = 0,


as a, _Lb. So (4.3) implies either P,,,q = 0 (wheneverf(b)p(a) # 0), which leads $0 q(al) = 0 by (I*), or f(b)p(u) = 0, which is equivalent (see (4.1)) to P,Pbf = 0. But on the other hand,

4(%) +4(c) = 4(a) = 13

and therefore the first possibility leads immediately to q(c) = 1, and thus by (I*>

P,q = q(c)r = r,

r being a pure state with r(c) = 1. Furthermore, as s(r) < c, one finds for the second possibility by (IV*) and (II*)

P,,,,q = P,(,,P,q = P,c,)r = r, (4.4)

(for the last equality in (4.4) see page 133), hence by using (I*) we find q(s(r)) = 1, which implies s(q) = s(r), hence q = r, as s is bijective (see Section 2), and thus finally

p,q = 4.

We therefore get for the first case (see (4.1)).

PcpuPbf = f(b)h)p,q = f(b)&)q = popbf,

but as c < a and c < b,

PcP, pbf = pcf

by (IV*) and (II*), and therefore

Pef = p, pbf,

as claimed. Consider now the second possibility. We then have, as f # 0 by the assumption,

0 = p.vpbf = f(b)p@)q = f(l)p/(b)i44q = f(l)p,(b, +c)P(% +c)q,

hence

(p,(b,)+p/(c))(p(a,)+p(c)) = 0.

Thus either pf(bJ+pf(c) = 0, which implies p,(b,) = p,(c) = 0, or p(ui)+p(c) = 0, hence p(u3 = p(c) = 0, which leads to s(p) < c’. But, at the same time, s(p) < b by (4.2), and thus s(p> < bl = b- c = b A c’, hence

But Pbf = f(b)p implieS P&,f = f(b)PQ, and, On the other hand, P&f = Pb,f, by (IV*) and (II*), as bl < b, and thus, after equating the two equations above, we find by using (I*)

f(b) = f(b)p(bd = f(b)(&Q)(l) = (Pblf )(I) = f(b,),

which implies f(c) = 0, as f(b) = f(b,) +f(c). Hence

Pr(C) = 0.

138 W. GU2

Thus, for both cases we have found the equality p,(c) = 0, which is equivalent tof(b) = 0, and therefore using (I*) one finds

PoAbf = Pcf = 0 = P&f.

The implication a c) b => P,, Pb = P, ,, b is thus proved. By symmetry (a t) b implies b c-) a) we have also shown that a t, b implies Pb P, = Ponb = P, Pb.

THEOREM 4.2. Let (Pa> be an arbitrary famiy of extended pure operations satisfying the Compatibility Postulate. Suppose a EL, a # 1, and let e be an arbitrary atom not con-

tained in a, e 4 a. Then there exists av e, av e-a is an atom, and

pave-*PC? = PdPt?? where, as usually,

pe: = s-‘(e).

Proof: To prove that ave exists, let us assume some c EL to satisfy c > a, e. Then the relations c > e, c tr a’ (see Section 2) imply, by (II*),

PCP, = Pc(P,Pe) = pep, = Pet

since P,p, = pe, as e = s(p,) (see page 133), hence’

PdP, = Pd pep, = PC PdPe

by Proposition 4.1. Hence, as by (I*)

Pdp, = p, Wp

for some pure state p with p(a’) = 1 (hence s(p) < a‘), one finds

p,(a’)p = p,(a’)P,p, hence

PCP = P, (4.5)

as p,(a’) # 0. (Indeed, p,(a’) = 0 leads to e = s(p,) 6 a, which contradicts OUT- assumption that e 4 a.)

From (4.5) we easily get p(c) = 1, hence s(p) < c. Note now that there exists av s(p), as s(p) _L a, and that avs(p) < c, as a < c and

s(p) d c. Since s(p)’ c--, a’ (see Section 2), one finds by Proposition 4.1

Ps~p~~ha~~e = Ps~,~Pdpe = pe(4Pswp = 0,

where the last equality follows from (I*), hence, again by (I*), pI (s(p)’ A a’) = 0, Which implies

e = s(pe) < (s(p)‘Aa’)’ = s(p)+a.

Collecting the inequalities

e < s(p)+a, a < @)+a,

s(p)+aGc= any upper bound for a and e,


we find that av e indeed exists, and

ave = &)+a.

Finally, as s(p) < a’, one finds by using (II*) and (IV*) that

or

pave-.Pc = P.YP,, as s(p) = av e-a by (4.6).

The theorem is thus proved.

(4.6)

COROLLARY ~Ks.~ Suppose (P,,} is a family of extended pure operations satisfying CF. If e is any atom 4 a’ (a E L, a # 0), then there exists eva’, eva’- a’ is an atom,

and P,Pt? = pc”o~-o~Pc, (4.7)

hence PczPt? = p,(a)pe V cr,-cr’ and E,,p, = peva~_.~.

Proof Only the equality P,p, = p,(a)pc vll,--o, (implying E.p, = P,,p,fp,(a) = pe V o.-d, as e .& a’ leads to p,(a) # 0) must be proved, but this follows directly from (4.7) by (I*) since

pe(eva’-a’) = p=(eva’)-pe(a’) = 1-p,(a’) = p,(a).

We thus have also established the following fact (see Corollary 3.7):

THEOREM 4.4. If the proposition-state system (L, S) satisfies all the conditions 1, 2, 3 of Section 2, then any .family {E4} f o normal g.p.0. satisfying the Compatibility Postulate

a c-, b =S E,Eb = EbE, (a, b E L, a, b # 0)

consists, in fact, of pure operations defined by the usual formula:

E,P, = peva*-a*.

This is the representation theorem for an axiomatically defined family of pure operations (normal, generalized-according to our terminology) satisfying the Compatibility Postulate.

5. Application to quantum axiomatics

Among all postulates assumed for the logic L the most controversial are the complete lattice property of L, the atomisticity of L, and the very covering law, as having no physical justification. Although the first two troubles connected with the complete lattice structure and the atomisticity of L are now resolved by a suitable embedding of the logic L (see, e.g., [l], [2], [6], [8]), the covering law was still left without a satisfactory physical justification for it. The results of the preceding section of this paper seem to give a satisfactory

* Under another axiom system a similar theorem was proved by Gunson (Commun. Math. Phys. 6 (1967). 262).

. 140 -,w. GUZ

answer to the problem of justifying of the covering law, as we have shown that the latter follows from the physically clear Compatibility Postulate, which connects the structure of the logic L with the structure of the “logic” of pure operations. Such a postulate seems to be necessary also from the mathematical point of view, as these two logics must be, clearly, related in a definite way.

Thus, if we complete the list of axioms for (L, 5’) formulated in [7] by adding to it the Compatibility Postulate, then we find the resulting axiom system to be both physically clear and sufficient to imply the well-known Piron-Mac Laren “Hilbert-space” representation theorem (see, e.g., [13], [Ill, [16]) for the logic L. The axioms there are (see [7]):

(1) L is an orthomodular a-orthoposet.

(2) The set S of states is a n-convex subset of the set of all probability measures on L.

(3) For each non-zero proposition a E L there exists a pure state p E P with p(a) = 1.

(4) If m(u) = m(b) = 1 for some m E S and a, b EL, then there is a proposition c < a, b such that m(c) = 1.

(5) L is separable, that is every subset of pairwise orthogonal propositions from L is at most countable.

(6) If for any a E L, for whichp(a) = 1, we have also q(u) = 1 (p, q being pure states), then p = q.

(7) L is irreducible.

(8) The Compatibility Postulate holds for L, i.e. there exists a family of normal g.p.0. associated with L, which satisfies CP.

Remark: The axiom (6) expresses the physically obvious fact that a single pure tate cannot produce any superposition other than itself (see [I], [5]).

It can be shown (see, e.g., [7]) that the axioms (l)-(6) imply the logic L to be a complete atomistic lattice possessing the property 3 from Section 2. By (8) it has also the covering property, as was shown in Section 4, and therefore our axioms (l)-(B) are sufficient to obtain the standard representation theorem for the logic L (see, e.g., [13], [ll], [16], [12]), which states the following:

For any irreducible atomistic complete orthocomplemented lattice L of length 2 4 with the Covering Law holding in it, there exists a division ring D with an involutive antiautomorphism *, a vector space Y over D, and a definite (non-degenerate) *-bilinear form ( ., *) on V such that L is orthoisomorphic to the ortholattice of all ( ., *)-closed linear manifolds in V.

REFERENCES

[I] Bugajska K., and S. Bugajski: Rep. Math. Phys. 4 (1973), 1. [2] -: Ann. znst. lx Poincare’ 19 (1973), 333. [3] -: Bull. Acad. Polon. Sci., Ser. math., 21 (1973), 873.


[4] GIJZ, W.: Rep. i$fQth. f'hys. 2 (1971), 53. [5] -: ibid. 6 (1974), 445. [6] -: ibid. 7 (1975), 313. [7] -: Znt. .Z. Theor. Whys., 16 (1977), 299. [8] -: Ann. Inst. H. Poincure’, 28 (1978), 1. [9] Jauch, J. M., and C. Piron: Helv. Phys. Actu 42 (1969), 842.

[lo] Mackey, G. W.: The mathematical foundations of quantum mechanics, Benjamin, New York, 1963. [ll] Mac Laren, M.D.: PaciJic J. Math. 14 (1964), 597. 1121 Maeda, F., and S. Maeda: Theory of symmetric lattices, Springer, Berlin-New York, 1970. [13] Piron, C.: Helv. Phys. Acta 37 (1964), 439. [14] Pool, J. C. T.: Commun. Math. Phys. 9 (1968), 118, 212. [15] Varadarajan, V. S., Commun. Pure Appl. Math. 15 (1962), 189; correction, ibid. 18 (1965). [16] -: Geometry of quantum theory, Vol. 1, Van Nostrand, Princeton, 1968. 1171 Zierler, N.: Pacific i’kfuth. 11 (1961), 1151.

pure operations and the covering law

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