Vol. 47 (2001) REPORTS ON MATHEMATICAL PHYSICS No. 2

COEXISTENCE VS. FUNCTIONAL COEXISTENCE OF QUANTUM OBSERVABLES

PJZKKA LAHTI

Department of Physics, University of Turku, FIN-20014 Turku, Finland (e-mail pekka.lahti@utu.fi)

and

SYLVIA PULMANNOVA

Mathematical Institute, Slovak Academy of Sciences, SK-814 73 Bratislava, Slovakia (e-mail: pulmann@mau.savba.sk)

(Received March 24, 2000)

We consider the question when coexistent observables are functionally coexistent and we give partial answers for special classes of observables on effect algebras, in particular on the Hilbert space effects.

Keywords: coexistent observables, joint observables, joint measurability, effect algebras.

1. Introduction

Let ‘FI be a complex Hilbert space, and let L(3-I) denote the set of bounded operators, E (Z) = (A E C(N) 1 0 5 A 5 I) the set of effects, and P(x) = (P E C(N) 1 P = P2 = P*} the set of projections on 7-k Let S2 be a nonempty set and A a a-algebra of subsets of R. Let E : A + ~C(ti) be an operator measure, which is positive, that is, E(X) 2 0 for all X E A, and normalized, that is, E(a) = I. We call such measures observubles.

Observables El : d1 --+ L(x) and E2 : A2 + C(‘H) are coexistent if there is an observable F : A +- 13(7-t!) such that ran(E1) U ran(E2) E ran(F), that is, for each X E .A1 and Y E d2. El(X) = F(Zx) and ET(Y) = F(Zy) for some 2x, 2~ E A. Observables El : d1 + C(‘FI) and E2 : d2 + L(B) are functionally coexistent if there is an observable F : A + C(R) and (measurable) functions fi : Sz + S2,

and f2 : S2 + C22 such that for each X E d1, Y E d2, El(X) = F(&-l(X)) and

E2( Y) = F(f2-’ (Y)). Functionally coexistent observables are coexistent, but we do not know whether these notions coincide. On the other hand, functionally coexistent observables can be characterized in terms of biobservables and joint observables 11, 21.

In this paper we shall investigate the question under which conditions coexistent observables are functionally coexistent. Our results do not depend on the special

11991

200 E? LAHTI and S. PULMANNOVA

properties of a Hilbert space. Therefore, we formulate and study this question in a more general frame of an effect algebra, treating, however, the important Hilbert space case separately in the final section of the paper.

A necessary and sufficient condition when for two real observables E, F on an effect algebra there is a measurable function f such that E = F o f-’ is given in [3, A.4.12 Theorem]. This condition is exactly what is needed to apply a procedure analogous to [4, Theorem 1.41. This condition is, however, rather general and hard to be checked. In the present paper, we present some classes of observables, clas- sified by properties of their ranges and the corresponding morphisms, which imply functional coexistence.

We start with a brief summary of the basic notions and definitions on effect algebras which shall be needed here. For more details and motivation, the reader is referred to [5-Q.

2. Basic definitions and notions

2.1. General

An e$ect algebra is a system (L, 0, 1, @) where L is a nonempty set, 0, 1 are distinct elements of L and $ is a partial binary operation on L that satisfies the following conditions (we write a _L b if a $ b is defined):

(El) If alb, then blu and b@u=u@b. (E2) If a _L b and c I (u@b), then b I c, a I (b@c) and u$(b$c) = (u@b)@c. (E3) For every a E L there exists a unique a’ E L such that a I u’ and a $ a’ = 1. (E4) If a I 1, then a = 0.

In the sequel, whenever we write a @ b we are implicitly assuming that a _L b, that is, the sum a $ b of a and b is defined in L.

We define a I b if there is a c E L such that a $ c = b. It can be shown that the element c is unique, and that (L, 5,’ ) is an ordered set with 0 5 a 5 1 for all a E L, u” = a and a 5 b implies b’ ( a’. This means, in particular, that de Morgan’s laws hold in (L, I,‘): for any a, b E L, a A b exists if and only if a’ v b’ exists, in which case (a A b)’ = a’ v b’. We note also that a I b if and only if a 5 b’. If a 5 b, that is, b = a $ c we also write c = b 8 a for the unique effect c E L. We call the elements of L effects.

We say that an effect a E L is sharp if the greatest lower bound of a and a’ exists in L (with respect to the order 5 defined by @) and equals with the least element 0 of L, that is, a A a’ = 0. An effect a E L is called regular if a $ a’ and a’ $ a. Clearly, a sharp effect is regular, but a regular effect need not be sharp.

A simple example of an effect algebra is the unit interval [0, l] G IR of the real line IR. For a, b E [0, l] we define u I b if a + b p 1 and in this case a @b = a + b. For a E [0, l] we have a’ = 1 - a.

A a-algebra A of subsets of a nonempty set !LZ constitutes another example of an effect algebra when X @ Y is defined whenever X rl Y = 0 with X @ Y = X U Y. Clearly, X’ = Q \ X and the order defined by @ on A coincides with the set

COEXISTENCE VS. FUNCTIONAL COEXISTENCE OF QUANTUM OBSERVABLES 201

inclusion. The order structure of the effect algebra (A, @, 0, R) is that of a Boolean a-algebra.

The set E (3-t) of the Hilbert space effects is a third important example of an effect algebra: for any A, B E E (3-1), if A + B E E (ti), we define A $ B = A + B. Again, it is immediate that A’ = I - A and that the order induced on & (3-1) is the usual order of self-adjoint operators which was used to define the set & (‘FI). The effect algebra (E (IFI), @, 0, I) does not form a lattice. An effect A E &(7-l) is sharp exactly when it is a projection, and it is regular whenever its specturm extends both below and above 4.

A sube$ect algebra of an effect algebra L is a subset K of L such that 0, 1 E K, a E K implies a’ E K and a, b E K, a I b, implies a $ b E K. Notice that a subeffect algebra is an effect algebra in its own right. Moreover, the ordering in K is that inherited from L. Indeed, denote by 5 K the ordering in K, that is, a 5~ b if there is c E K such that a @ c = b. Clearly, if a 5~ b then a 5 b in L. Conversely, assume that for some a, b E K we have a 5 b. Then there is c E L with a $ c = b. Now, since b’ @ a @ c = 1, we get c = (b’ @ a)’ E K. Therefore a 5~ b. The effect algebra (A, $, 0, Q) is, in a natural way, a subeffect algebra of (2”, @, 0, S2) (with the same definition of @). The set P(N) of all projection operators on a Hilbert space 7-1 is a subeffect algebra of E (‘H), the sharp elements of & (8). It is known to be a complete orthomodular lattice.

2.2. Morphisms and observables

If K and L are effect algebras, we say that 4 : K + L is additive if a _L b implies #(a) I 4 (b) and #(a @K b) = @(a) $L q5 (b). If 4 : K + L is additive and @( 1 K) = 1 L, then 4 is a morphism. If 4 : K + L is a morphism and $(a) I 4(b) implies a _L b, then 4 is a monomorphism. A surjective monomorphism is an isomorphism. It is easy to see that $J is an isomorphism if and only if 4 is bijective and 4-l is a morphism. We note that a morphism $J : K + L preserves the order and the ‘complement’. Indeed, for any a E K, @(lo) = $(a @K a’) = @(a) @IL ~$(a’) = 1~ showing that $~(a’) = @(a)‘. Also, if a 5~ b, then b = a @K c for a unique c E K, so that 4(b) = #(a) @L 4(c), which means that $(a) IL 4(b). For a, b E K, if a AK b exists in K, then also #(a AK b) (L ~+(u),@(b), but @(a AK b) need not be the greatest lower bound of #(a), 4(b) even if $(a) AL 4(b) would exist in L. A morphism @J is called a A-morphism if $(a) AL 4(b) exists whenever a AK b exists, and $(a AK b) = @(a) ~~ d(b). A v-morphism is defined dually. By de Morgan’s laws a A-morphism is also a v-morphism, and vice versa.

An effect algebra is called a a-e$ect algebra if for any increasing sequence ui 5 a2 p ... of effects the supremum V ui exists in L. Equivalently, L is a a-effect algebra if for any summable sequence (Ui)ieN (which is a sequence such that @icnUi :=UI @Q@...

VII @i-L

CD a, exists for any n E N), the element BiEw ui := ai exists in L. The above examples of effect algebras all are a-effect

algebras. A morphism #J : K + L, where K is a a-effect algebra, is a a-morphism if #(ei,,ai) = ~i,,9(Ui) for any summable sequence (ai)icw. Similary, one defines the notions of 0 - v and 0 - A-morphisms.

202 P. LAHTI and S. PULMANNOVA

Let L be an effect algebra and (R, A) a measurable space. A a-morphism E : A + L is called an observable. In other words, a map E : A + L is an observable if

(01) E(a) = 1~9 (02) E(Ui,MXi) = BicN E(Xi) for any disjoint sequence (Xi) C A, meaning that

the o-sum eicN E(Xi) exists in L and equals with E(Ui,wXi).

The following observations are immediate.

LEMMA 2.1. Let E : A + L be an observable. Then (a) for any X E A, E(X’) = E(X)‘; in particular, E(0) = 0~; (b) if X, Y E d, X G Y, then E(X) 5 E(Y); in fact, E(Y) = E(X) $ E(Y \ X),

that is, E(Y \ X) = E(Y) 8 E(X); (c)for each X,Y Ed, E(XUY)=E(X\(XnY))@E(Y\(XnY))@E(XfIY),

that is, E(X u Y) 8 E(X n Y) = (E(X) 8 E(X fl Y)) @ (E(Y) 8 E(X n Y)).

We stress that in the effect algebra A, if (Xi)iew c A is a disjoint sequence, then eiGN Xi = Ui,NXi = VicN Xi, whereas in L the existence of @ieN E (Xi) does not imply the existence of View E(X;) in L. Therefore, an observable E : A + L need not be a (T - v-homomorphism. Also, though E(X n X’) = OL, the quantity E(X) ~~ E(X)’ need not exist at all, and if it does, it need not be equal to the zero effect 0~. Similarly, E(X n Y) may or may not be equal to E(X) AL E(Y), even if the latter quantity exists, at all. We call the space (S’Z, A) the value space of the observable. We consider the case (1;2, d) C (R”, B(IR”)), and we say that E is a real observable if it is defined on the real Bore1 space (R, f?(R)). Observables El : dt + L and Ez : AZ + L are coexistent if there is an observable F : A -+ L such that

cc> ran(E1) U ran(E2) & ran(F),

and they are functionally coexistent if there is an observable F and measurable functions ft :a+Qt and fz:Q+Qz such that

(FC) El = Fo f,-‘, E2= Fo &-I.

3. ‘J%vo-valued observables

We say that an observable E is two-valued if its range is of the form ran(E) = {O,a,a’, 1) for some effect a E L.

PROPOSITION 3.1. Any two coexistent two-valued observables (0, a, a’, 1) and (0, b, b’, l} are functionally coexistent.

Proof: To demonstrate this simple fact, let (cz, a’] and {/I, /?‘) be two point value sets of the observables (0, a, a’, 1) and {0, b, b’, l), respectively, and let F be an observable such that F(X) = a and F(Y) = b. Consider the partition R = (X fl Y, X’ fl Y, X n Y’, X’ n Y’} of the value space Q of F into disjoint d-sets,

COEXISTENCE VS.FUNCTIONALCOEXISTENCE OFQUANTUM OBSERVABLES 203

and let 1 H F(X n Y), 2 H F(X’ II Y), 3 H F(X n Y’), 4 H F(X’ n Y’), constitute a corresponding coarse-grained observable FR of F. The maps ft : 1,3 I-+ a; 2,4 H o’, and f2 : 1,2 H /I; 3,4 t-k p’, allow one to write u = FR(fi_‘(cx)) = F(XnY)@F(XnY’) and b = FR(f2-l(B)) = F(XnY)@F(X’nY), showing that the two-valued observables are, indeed, functionally coexistent. 0

4. Sharp and regular ohservahles

An observable E is called sharp if its range ran(E) consists of sharp effects and it is called regular if the only irregular elements in its range ran(E) := {E(X) : X E A} are 0 and 1. Evidently, a sharp observable is regular but the converse is not true.

THEOREM 4.1. Let E : A + L be a regular observable. Then ran(E) is a a-subeffect algebra of L. Moreover, with respect to the partial order p inherited from L, ran(E) is a Boolean algebra and E is a o - v (and o - A)-morphism.

Proof: In the first part of the proof we follow [6]. Let a, b E ran(E) and a I b. We have to prove that a $ b E ran(E). Assume a = E(X), b = E(Y). Then X = X fl Y U (X \ (X n Y)), Y = X n Y U (Y \ (X fl Y)). Hence E(X fI Y) 5 E(X) = a, E(X fl Y) 5 E(Y) = b. Since a I b, that is, b 5 a’, it follows that E(X rl Y) ( E(X fl Y)‘, so that by the regularity assumption, E(X fl Y) = OL. Therefore a = E(X1) and b = E(Yl), where Xt := X \ (X n Y), Yt := Y \ (X fl Y) are disjoint sets. So we get a $ b = E(X,) $ E(Y1) = E(X, U Y,) E ran(E). We note also that X n Yt = 0 and a @ b = E(X U Yl). This observation will be used in the next paragraph.

Now assume that (ai)ieN is a summable sequence in ran(E). Using the above argument, we find disjoint sets X1, X2 such that al = E(Xl), a2 = E(X2). Now we proceed by induction. Assume that we have already found disjoint sets Xt, . . . , X,-l such that ai = E(Xi), i = 1,. . .,n-1. Then al@.=.$a,_l = E(XlUX2U.. .UX,_l). By the summability assumption (al @ . . - $ a,_l) I a,,. Therefore, there is a set X, E A such that (X 1 U . . . U X,-l) n X, = 0, and a, = E(X,). Thus we find a sequence Xi, i E IV, of disjoint sets such that at = E(Xi), i E N. From the a-additivity of E we obtain E(U, Xi) = ei E(Xi) = ei ai, which shows that ran(E) is closed under sums of summable sequences. It follows that ran(E) is a a-subeffect algebra of L.

Let E(X), E(Y) E ran(E). We will prove that E(X n Y) = E(X) A,,(E) E(Y), that is, E : A + ran(E) is a r\-morphism. Evidently, E(X n Y) 5 E(X), E(Y). Assume that for some 2 E A, E(Z) i: E(X), E(Y). We can write 2 = (Z n X n Y) U (Z n (X n Y)‘). Moreover,

E(Z n (X n Y)‘) = E(Z n (X’ U Y’))

= E((Z rl X’ fl Y) U (Z n X’ n Y’) U (Z fl X fl Y’))

= E(Z n X’ n Y) @ E((Z tl X’ n Y’) @ E((Z fl X fl Y’))

i E(Z) i E(X), E(Y).

204 F! LAHTI and SPULMANNOVA

But we also have E(ZflX’nY) 5 E(X’), E(ZnXnY’) _( E(Y’), E(ZnX’nY’) p E(X’), E(Y’), so that the effects E(Z n X’n Y), E(Z n X’n Y’), and E(Z n X n Y’) are irregular and thus equal 0 L. Therefore also E(Zfl(XrlY)‘) = 0~. Thus E(Z) = E(Z fl X fl Y) _( E(X fl Y). This concludes the proof that E(X fl Y) = E(X) A,,(~) E(Y). By de Morgan’s laws one gets the dual result: for any X, Y E A, X n Y = 0, E(X U Y) = E(X) @ E(Y) = E(X) vran(~) E(Y). Also, if (Xi) c A is a disjoint sequence, then

E(UEIXi) = & E(Xi) = v(&E(Xi)) i=l n=l i=l

= V b’ran(~,(E(X~), . . ., E(Xz)D n=l

= vran(~)IE(&) I i E W.

To prove that ran(E) is a Boolean algebra, it remains to prove distributivity. This follows immediately from the fact that E is a A-morphism and a v-morphism from a Boolean set. cl

COROLLARY 4.2. The range ran (E) of an observable E is a Boolean o-algebra (with the ordering inherited from L) if and only if E is regular.

Proof: We have to prove the ‘only if’ part. Hence, assume that ran(E) is Boolean, and let a = E(X) be an irregular element. Then a _L a, that is, E(X) 5 E(X)‘, which in a Boolean algebra implies that E(X) = a = 0. 0

The classical results of Sikorski [9] and Varadarajan [4] lead now to the fol- lowing conclusion.

COROLLARY 4.3. Any two observables whose ranges are contained in the range of a regular observable are functionally coexistent.

We close this section with a further characterization of regular observables. We note that it was shown in [3, Appendix A] that property (ii) in the next theorem implies V-property (see next section). Our result is stronger.

THEOREM 4.4. Let E : A + L be an observable. The following properties are equivalent:

(i) E is regular, (ii) for any X, Y E d, E(X) 5 E(Y) implies E(Y) = E(X U Y).

Proof: (i) + (ii). Assume that E is regular, and E(X) 5 E(Y). Then E(X U

Y) = E(X) V,“(E) E(Y) = E(Y). (ii) + (i). Assume next that E(X) 5 E(X’). Then, by (ii), E(X’) = E(XUX’) =

1 L, so that E(X) = OL, which shows that E is regular. 0

COEXISTENCE VS.FlJNCTIONALCOEXISTENCEOF QUANTUM OBSERVABLES 205

5. V-property and strong observables

Let E : A + L be an observable and Q a subset of ran(E). Following Chovanec and Kopka [3, Appendix A] we say that E has the V-property on Q if the following condition is satisfied:

(V) for any X, Y E A, X G Y, and c E Q, the inequalities E(X) 5 c 5 E(Y) imply that there is a Z E A such that X E Z E Y and c = E(Z).

The purpose of introducing this notion in [3] was the following theorem, which can be proved by a suitable adaptation of the well-known theorem of Varadarajan [4, Theorem 1.41. For completeness, we sketch here the proof.

THEOREM 5.1. Let E, F be real observables on an efect algebra L such that ran(E) E ran(F), and let F have the V-property on ran(E). Then there is a Bore1 function f : R + lR such that E(X) = F o f-‘(X), X E B(R).

Proof: By property V, if X, Y E a(R) , X S Y and F(X) p c I F(Y) for some c E ran(E), then there is a Z E f?(R) such that F(Z) = c and X G Z G Y. Let rl,r2, . . . be any distinct enumeration of rational numbers and let Dj := {t E IR 1 t 5 ri }. Evidently, E(Q) 5 E (Dj) whenever r; 5 rj. We shall now construct sets Zi,Zz,... in E(R) such that (a) F(Zi) = E(Di), i = 1,2,3,. . . , (b) Zi E Zj whenever ri 5 rj.

Let Z1 E E(R) be such that F(Z1) = E(Q). Suppose Zi, Z2,. . . , Z, E E(R) have been constructed such that (i) F(Zi) = E(Di) for i = 1,2,. . . , n, (ii) Zi C Zj whenever ri 5 rj, 1 5 i, j 5 n. We shall construct Zn+l as follows. Let (il, i2, . . . , i,) be the permutation of (1,2, . . . , n) such that ri, 5 ri2 5 . . . 5 ri,. Then there exists a unique k such that rik ( r,+l 5 Tit+, (we define ri,, =

-00, Tin+, = +oo), and by property V, we can select Zn+l such that Zi, G Zn+t c

zik+, * The collection (Zt , Z2, . . . , Z,+l ) then has the same properties relative to rl,r2,..., r,+l as (Zl,Z2,..., Z,,) had relative to r-1, t-2, . . . , r,. It follows by in- duction that there exists a sequence Z1, Z2, . . . in Z?(R) with properties (a) and (b). Since F(nZi) 5 E((-co, r-i]) for all ri, F(nZi) = 0~. Now replacing Zk by Zk \ nj Zj if necessary, we may assume that ni Zi = 0.

Now the proof follows exactly the pattern of [4, Theorem 1.41. 0

For example, let E : (1,2,3,4) + [0, l] be an observable defined by E(i) = l/4, i = 1,2,3,4, and F : (1,2,3) + [0, l] be an observable defined by F( 1) = l/2, F(2) = F(3) = l/4. Then ran(E) = ran(F) as sets, and E has the V-property on the range ran(F) of F, but F does not have the V-property on the range ran(E) of E. Indeed, (2) G (1,2} and F(2) = l/4 5 l/2 5 3/4 = F(1,2), but there is no X, {2} E X E {1,2) with F(X) = l/2. Define f : (1,2,3,4} + (1,2,3} by f(1) = 1 = f(2), f(3) = 3, f(4) = 4. Then F = E o f-l. There is no function g such that E = F o g-l.

We can give another characterization of V-property using congruences and strong morphisms of effect algebras as introduced in [lo] and [ 111. We start with recalling some definitions.

206 I? LAHTI and S. PULMANNOVA

Let L be an effect algebra. A binary relation - on L is called a congruence [lo] if:

(Cl) - is an equivalence relation,

(C2) a--t, b-b,, alb, al Ibi imply a@b-ut$bt, (C3) a I b and b - c imply that there is d E L such that d - a and d I c.

If - is a congruence we let [a] denote the equivalence class of a E L. Define [a] I [b] if there are al E [a] and bl E [b] such that at I bl, and then define [a] @ [b] = [al G3 bl]. It has been proved in [l l] that the quotient space P/- := [[a] 1 a E P}, with the operation $ defined above and with [0], [l] as zero and unit, is an effect algebra.

Let K and L be effect algebras. We say that a morphism h : K -+ L is strong if it has the property

(S) for a, b E K, if h(u) _L h(b), then there is ai E K such that ai I b and h(ul) = h(u).

For the proof of the following result, see [ 10, 111.

LEMMA 5.2. Let K, L be effect algebras, h : K + L a morphism. The following are equivalent:

(i) h is a strong morphism, (ii) the relation a - b, dejned by h(u) = h(b), is a congruence.

THEOREM 5.3. For any observable E : A + L the following statements are equivalent:

(i) E has the V-property on its range ran(E), (ii) E is a strong morphism.

Proof: Let E be a strong morphism. Assume X, Y, Z E A, X C Y and E(X) ( E(Z) 5 E(Y). Since E(Z) 5 E(Y), that is, E(Z) I E(Y’), by the strong property of E, there is a Zi I Y’, that is Zi c Y, with E(Z1) = E(Z). It follows that

E(Y) 0 E(Z) = E(Y \ ZI) E ran(E).

Moreover,

E(Y \ Zl) = E(Y) 8 E(Z) 5 E(Y) 8 E(X) = E(Y \ X).

Therefore, by the strong property, there is VI G Y \ X such that E (Y \ Zl ) = E ( VI). From E(Y) 8 E(Z) = E(Vl) we get

E(Y \ VI) = E(Y) 8 E(Vl) = E(Z).

Now VI G Y \ X implies X S Y \ VI G Y. This proves the V-property on ran(E). Conversely, assume that E has the V-property on ran(E). Let E(X) 5 E(Y),

then E(0) ( E(X) ( E(Y) and 0 G Y, and E(X) E ran(E). By property V, there is Xi 2 Y such that E(X,) = E(X), and the strong property follows. 0

We say that an observable E is strong if E is a strong morphism.

COEXISTENCEVSFUNCTIONAL COEXISTENCEOFQUANTUM OBSERVABLES 207

COROLLARY 5.4. Any two observables whose ranges are contained in the range of a strong observable are functionally coexistent.

6. The Riesz decomposition property

Consider an effect algebra (15, @, 0, 1). We say that L has the Riesz decompo- sition property [ 121 if the following is true in L,

(R) for any al, a2, bl, b2 E L, if al @ a2 = bl $ b2, there are elements wij, i = 1,2, in L such that al= ~11 ~3 ~12, a2 = ~21 $ ~22, bl = ~11 $ ~21 and b2 = ~12 eB ~22.

The Riesz decomposition property can be extended by induction to any m, n E N, so that if ai,i=l,2 ,..., m, and bj,j=l,2 ,..., n, are such that

al G3.a. @a,, = bl @. . .@ b,,

then there are wij, i 5 m, j ( n, such that

ai = @jwij, bj = @iwij.

THEOREM 6.1. Let E be a strong observable on an e$ect algebra L (equivalently, E has the V-property on ran(E)). Then ran(E) is a subeffect algebra of L, which has the Riesz decomposition property.

Proof: To show that ran(E) is a subeffect algebra of L, let a, b E ran(E) be such that a G3 b is defined in L. Since E is strong, we can find X, Y E A such that X n Y = 0 and a = E(X), b = E(Y). Then a Cl3 b = E(X U Y) E ran(E). With Lemma 2.1 this proves that ran(E) is a subeffect algebra of L.

Now we prove the Riesz decomposition property. Assume that the effects al, a2, bl, b2 E ran(E) are such that al $a2 = bl @b2. Since E is strong, we find a disjoint pair Yt, Y2 in A such that bl = E(Yl), b2 = E(Y2). Now al $ a2 = bl $ b2 = E ( YI U Y2). From al 5 E (Y1 U Y2), using once more the strong property, we can find X1 such that al = E(Xl) and Xt C Yt U Y2. Put X2 := (Yl U Y2) \Xt. Then we have XI U X2 = Yl U Y2, E(X1) $ E(X2) = E(YI) CD E(Y2). Put wll = E(XI n YI), 2~12 = E(Xl \ (Xl fl Yl)), ~21 = E(Yl \ (Xl fl Yl)). Now we have

E(X1) @ w21 = EGl u Vl \ (Xl n Yl)))

= EVl U (XI \ (Xl n YI))>

= E(YI) CB ~12.

Since X1 c XtlJX2, Yt E YtUY2, and XtUX2 = YlUY2, we have XtUYt G YtUY2, so that E(Xt)@w21 = E(Yt)@wl2 I E(YlUY2). Rut ~22 = E(YlUY&3(E(Xl)@wz1) = E(YI U Yz) 8 (E(Yt) @ ~12). Then

E(X1) @ E(X2) = ~11 CB w12 @ w21@ w22

= EVl) CB E(Y2),

hence EGG) = ~21 CD ~22, E(Y2) = w12 $ ~22. Cl

208 F? LAHTIand S.PULMANNOVdr

We say that an observable E : A + L is simple if there is a finite subset 520 of SJ such that E(C20) = 1.

~OPOSITION 6.2. Any two simple observables E, F on an effect algebra with the Riesz decomposition property are jiuzctionally coexistent.

Proof: Let (hi,. . . ,A,}, {PI,. . . , p,) be finite sets such that E({hi)) = ai, F((pj}) = bj, and @iirnai = 1~ = @jsmbj. Then there are “ii with ai = $jwij, bj = $iwij. Clearly, @ijwij = 1 L,, and we may define a simple observable G with value space {wij} such that G(~ij) = wij. NOW define functions e : (wij} + (Ai},

f : {oij} + (pj} such that e(wij) = J.i, f(wij) = pj. Then E = G o e-‘, F =

G o f-l. 0

F&MARK 6.3. In [l, Section 2.51 we claimed that any two simple observables are functionally coexistent. In the above proposition this claim is proved under the additional assumption on the Riesz decomposition property. We do not have any counterexample to disprove the more general statement of [l]. However, as pointed out by Dr. A. DvureEenskij, the construction of the function fi, resp. f2, in [l] may fail since the involved sets Xi, resp. Yj, need not be disjoint.

7. Regular vs. strong observables

A comparison between regular and strong observables can be derived from the following propositions.

PROPOSITION 7.1. Let E : A + L be an observable. The following statements are equivalent:

(i) E : A + L is regular, (ii) X - Y if E(X) = E(Y), X, Y E d coincides with the congruence on A

generated by the ideal Z, := {X E A 1 E(X) = 0).

Proof: (i) =F (ii). By Theorem 4.1, ran(E) is a subeffect algebra of L, which is a Boolean algebra with respect to v and A taken in ran(E). Moreover, the mapping E : A + ran(E) is a Boolean algebra (a-) morphism. Therefore ZE is an ideal in d, and the quotient d/ZE is isomorphic with ran(E) as Boolean algebras. Indeed, we recall that the Boolean algebra congruences coincide with the effect algebra congruences (cf. [lo]), and X -lE Y iff (X\XnY)U(Y\XnY) E ZE. Now, E(X) = E(Y) implies (X \ X n Y) U (Y \ X fl Y) E ZE, that is, X -I~ Y. Conversely, if X -lE Y, then E(X) = E(X n Y) = E(Y). Therefore - = rvlE.

(ii) =9 (i). (“) ’ pl’ u im ies that ran(E) is a Boolean algebra, hence by Corollary 4.2, E is regular. 0

COROLLARY 7.2. Every regular observable is strong. The converse does not hold.

Proof: The statement that regular implies strong can be derived from Proposition 7.1 and Lemma 5.2. Alternatively, we can use Theorem 5.3 and show directly that condition (ii) of Theorem 4.4 implies the V-property.

COEXISTENCE VSFUNCTIONAL COEXISTENCEOF QUANTUM OBSERVABLES 209

As a counterexample, consider the observable E : { 1,2,3,4) + [0, l] defined by E(i) = l/4, i = 1,2,3,4. It can be easily checked that E is a strong morphism.0

8. The Hilbert space case. Projection valued observables

We recall that in the Hilbert space frame an observable E : A + E (‘FI) is regular if for each X E A, such that 0 # E(X) # I,

E(X) $ ;Z, ; i.? E(X).

With this specification all the results from the preceding sections can be applied. The sharp elements of E (‘FI) coincide with the projections and they form a subeffect algebra P(x) of E (3-t), which is an orthomodular lattice. Moreover, the infimum of any two projections exists in & (‘H), and is again a projection. Thus the lattice operations in P(7-l) coincide with those in & (‘H).

An observable E : A + E (ti) is sharp if it is projection valued, that is, if E(X)* = E(X) for all X E A. It is well known that this is the case exactly when E is multiplicative: E(X fl Y) = E(X)E(Y) for all X, Y E A. For a Hilbert space observable E : A + E (3-t) Lemma 2.1 gets the familiar form:

l for any X E A, E(X’) = I - E(X), a if X, Y E A, and X c Y, then E(Y \ X) = E(Y) - E(X), l for all X, Y E A, F(X U Y) + F(X n Y) = F(X) + F(Y).

Moreover, for any A E E (‘H), P E P(li), if A 5 P, then A = AP = PA. The just quoted results lead immediately to the following known result.

LEMMA 8.1. For any observable E : A + E(x), if E(X)* = E(X) for some X E d then E(X)E(Y) = E(Y)E(X) for all Y E d

Proof: Assume that E(X)* = E(X) for some X E A. For any Y E A, Xn Y C Y, so that E(Y) = E(Y \ (X n Y)) + E(X fl Y), and E(X) + E(Y) - E(X rl Y) = E(X) + E(Y \ (X n Y)) = E(X U Y) 5 I. Therefore, the effects E(X fl Y) and E(Y \ (X n Y)) are below the projections E(X) and Z - E(X), respectively, so that

E(X n Y) = E(X)E(X n Y)E(X),

E(Y \ (Y n X)) = (I - E(X))E(Y \ (Y fl X))(Z - E(X).

Therefore, E(Y) = E(Y \ (X n Y)) + E(X n Y) = E(X)E(X n Y)E(X) + (I - E(X))E(Y \ (Y rl X))(Z - E(X), which gives through multiplication by E(X) that E(X)E(Y) = E(Y)E(X). 0

As an immediate corollary to this lemma one obtains the following corollary.

COROLLARY 8.2. Let El and E2 be two coexistent observables. Zf one of them is projection valued, then they are functionally coexistent.

Proof: Assume that El is projection valued. Since El and E2 are coexistent, Lemma 8.1 implies that El and E2 are commuting: E1(X)E2(Y) = Ez(Y)El(X)

210 P. LAHTI and KPULMANNOVA

for all X E Al, Y E AZ. Thus the map dt x A2 3 (X, Y) I-+ El(X)Ez(Y) E &(7-t) determines a biobservable of El and E2, which is to say that El and E:! are functionally coexistent. q

COROLLARY 8.3. Let El : At + E (7-L) and E2 : d2 + E (7f) be coexistent projection valued observables and let F : A + & (7-t) be an observable such that ran(Ei) U ran(E2) S ran(F), so that, for any X E At, El(X) = F(Zx), for some Zx E d, and for any Y E d2, E2( Y) = F(Zy) for some Zy E A. Then

F(Zx n ZY) = EI(X) ~~(7-r) E2UY (1)

= EI GO fvq7-1) E207

= El W)E2W,

F(Zx U ZY) = EIW) VE(H) E2W (2)

= EIGO ~~(7-1) E2W

= El (Xl + E,(Y) - EI GOE2(Y),

El(X’) = F(Zxt) = F(Z!J, (3)

E,(Y’) = F(Zy,) = F(Z;), (4)

F(Z;, U Z;,) = F(Zx U Z,). (5)

Proof: For any two projections P, R E P(‘H) c E (1-1) their greatest lower bound in E (fl) exists and equals with their greatest lower bound in P(x): P AE (7-1) R = P A~(~) R. Since El(X) and E2(Y) are commuting projections, one now has

ElW)E2V) = El(X) AF(R) E2U’) = EIW) h(7-1) E2V>, so that F@x n ZY) i El(X)Ez(Y). But one also has F(ZinZy) 5 F(Zk) = F(Zx)* = El(X)‘- = El(X’) as well as F(ZinZy) 5 F(Zy) = Ez(Y), so that F(ZinZy) 5 El(X)‘E,(Y). Sim- ilarly, F(ZxnZb) i El(X)E,(Y)‘-, and F(Z;nZ&) i El(X)‘E2(Y)‘. But then Z = F((zxnz,)u(z;nzy)u(zxnz;)u(z;,nz~)) = F(zxnzy)+F(z;,nzy)+F(zxn Zf,>+F(Z;nZ;) i EI(X)E~(Y)+E,(X)*E~(Y)+E,(X)E~(Y)’+E~(X)’E~(Y)’ = I, which shows that Eq. (1) holds. In passing, we also saw that (3) and (4) are true. To demonstrate (2), we note that F(Zx U Zr) = F(Zx) + F(Zy) - F(Zx n Zy) = El(X) + Ez(Y) - E,(X)Ez(Y) = El(X) v E2(Y) = F(Zx) v F(Zy). Fi- nally, F(Zk, U Zk,) = F((Zx/ n Z,!)‘) = F(Zxl fl Zyl)’ = (El(X’) A E2(Y’))’ = E,(X) v E2(Y) = F(Zx U Zy). 0

COROLLARY 8.4. Zf El, E2 are real observable satisfying the conditions of Corol- lary 8.3, the Boolean u-algebra 13 generated by ran(Ei) U ran(E2) is contained in ran(F), and there is a sharp observable G such that ran(G) = t? (so that El = G o f,-’ and E2 = G o f2-’ for some measurable functions f,, f2).

Proof: According to Neumark’s theorem, there is a Hilbert space 6 contaning 7-L as a subspace, a projection M : ii + ‘FL and a projection valued observable fi on fi such that for any Z E A, F(Z) = MF(Z)M. Now for any X E Al, Y E dz, El(X) = F(Zx) = MF(Zx)M, E,(Y) = F(Zy) = MF(Zy)M. For any projection

COEXISTENCE VS. FUNCTIONAL COEXISTENCE OF QUANTUM OBSERVABLES 211

A on 31, M;1M is a projection iff AM = MA. So for any X, Y, F(Zx), F(Zr) commute with M, and therefore also the projections from the Boolean sub-a-algebra fi of P(g) commute with M. Moreover, B is contained in the Boolean a-algebra

ran F. Therefore M8’M is a Boolean a-algebra of projections on 7-1, which contains ran(Et) Uran(E2). The rest follows by [4]. 0

8.1. Examples

1. Consider a simple observable E : i H Ei E E (B), i = 1, . . . , m, which is commutative, that is, Ei Ej = Ej Ei. According to [ 131 there is an II x m stochastic matrix (hii), that is 0 5 hij 5 1 for all i, j and cycl hij = 1 for each i = 1, . . . , n,

and there are pairwise orthogonal projections Pi, . . . , P,, such that Cy=, Pi = I, and E is given by j H Ej := Cyc’=l hii Pi. In [14] this representation was used to give an example of a nonregular observable.

Consider the case m = n = 2. It has been shown in [14] that E is nonregular iff )cii I hi2 and h2i 5 )c22, in which case El 5 E2 = I - El. We note that in this case, E is also not sharp, unless the above inequalities are equalities, i.e., El = E2.

Consider next the case 12 = 2, m = 3. It has been shown in [14] that if El, E2, E3 are disjoint atoms in ran(E), the range of E cannot be Boolean. It follows that E cannot be regular. Notice that if hij = l/3, i.e., El = Ez = Es, then E is nonregular, but it is sharp.

2. Let 4, II/ be a pair of orthogonal unit vectors in C2. Consider the projection

operators P[$], P[@], and P, := P[l/&@ +i’@)], with r = 0, 1, 2,3. Let hi, i = 1,2,3, be any positive numbers which sum up to 1. Then i H Ei, with El = AlP[$], E2 = hlP[+], E3 = hzPa, E4 = )c2P2, Es = h3P1, E6 = hsP3, defines an observable E with a 64-element noncommutative range [ 141. If all the hi’s are nonzero, this observable is nonregular and nonstrong. Indeed, at least one of $s must be less than l/3. Let hi ( l/3, and j be such that the projections in Ei and Ej are orthogonal. Then Ei + Ei,j 5 I, but there is no subset A of (1,2,3,4,5,6), Afl (i, j} = 0 with E(A) = Ei. Nevertheless, it is easy to check that any observable F with ran(F) G ran(E) is a function of E.

9. Concluding remarks

There are following classes of observables (represented as positive normalized operator measures E : A + L(N)) which differ by properties of the corresponding a-morphism from a Boolean a-algebra A to the effect algebra of Hilbert space effects & (a):

1. Sharp observubles. The range of such an observable consists of projections (sharp effects), and is a sub-a-algebra of P(N), and a sub-effect algebra of & (NH). The corresponding morphism preserves countable suprema and infima, and suprema and infima with respect to the range coincide with suprema and infima in & (ti). Two coexistent sharp observables are always functionally coexistent. Also, any two observables whose ranges are contained in a range of a sharp observable, are func- tionally coexistent.

212 I? LAHTI and S. PULMANNOVA

2. Regular observables. The range of a regular observable consists of regular effects, and is a sub-effect algebra of & (ti). The corresponding morphism preserves countable suprema and infima with respect to the range, which may differ from suprema and infima in E (3-t), even when they would exist. Moreover, the range of an observable is a Boolean algebra with respect to the ordering inherited from E (‘H) iff the observable is regular. Any two observables whose ranges are contained in the range of a regular observable, are functionally coexistent.

3. Strong observables. The range of such an observable is a subeffect algebra of & (‘FI). The corresponding morphism has the following property: if E(X) + E( Y) 5 I for some X, Y E A then there is X1 E A such that X1 rl Y = 0 and E(X) = E(Xl). Suprema and infima may be not preserved, and the range need not be Boolean. But any two observables contained in the range of a strong observable are functionally coexistent.

4. The Riesz decomposition property. The range of an observable E satisfies the Riesz decomposition property if property (R) from Section 6 is satisfied with L replaced by ran(E). In this case, any two simple observables with ranges contained in ran(E) are functionally coexistent. In particular, the range of a strong observable satisfies the Riesz decomposition property.

Acknowledgements

The paper was written during the stay of S. Pulmannovh at the University in Turku. Work was supported by the Academy of Finland and grant VEGP 2/7193/2OOO of the Slovak Academy of Sciences.

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