Vol. 55 (2005) REPORTS ON MATHEMATICAL PHYSICS No. 2

NOTES ON COARSE GRAININGS AND FUNCTIONS OF OBSERVABLES

ANATOLIJ D V U R E C E N S K I J

Mathematical Institute, Slovak Academy of Sciences, SK-81473 Bratislava, Slovakia (e-mail: dvurecen @ mat.savba.sk)

P E K K A L A H T I

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

SYLVIA P U L M A N N O V A

Mathematical Institute, Slovak Academy of Sciences, SK-81473 Bratislava, Slovakia (e-mail: pulmann @ mat.savba.sk)

and

K A R l Y L I N E N

Department of Mathematics, University of Turku, FIN-20014 Turku, Finland (e-mail: kari.ylinen@utu.fi)

(Received October 21, 2004)

Using the Naimark dilation theory we investigate the question under what conditions an observable which is a coarse graining of another observable is a function of it. To this end, conditions for the separability and for the Boolean structure of an observable are given.

Keywords: semispectral measure, Naimark dilation, coarse graining, separable observable, Boolean observable.

1. Introduction

Let (f2, .A) be a measurable space, 7-[ a complex Hilbert space, /2(7-/) the set of bounded operators on ~ , and E • A --+ Z;(7-/) a normalized positive operator measure, that is, a semispectral measure. We call such measures observables of a physical sysNm described by 7-/.

Let (/~, E, V) be a Naimark dilation of E into a spectral measure /~, that is, ffS • A --+/2(/~) is a projection measure acting on a Hilbert space /~ and V : 7-/--+/~ an isometric linear map such that E(X) = V * E ( X ) V for all X ~ .d. We say that an observable E : .A --~ l;(7-{) is separable if i t has a Naimark dilation (/~, E, V) which is separable, that is, the range E(.A) of E is a separable Boolean sub-o--algebra in the projection lattice T~(/C) of the Hilbert space /~. (We use the lattice theoretical terminology as introduced in [13].)

[241]

242 A. DVURECENSKIJ, R LAHTI, S. PULMANNOVA and K. YLINEN

We recall that a Boolean sub-o--algebra /3 of 79(E) is separable, if there exists a countable subset B such that the smallest Boolean sub-a-algebra of /3 containing B is /3. The importance of such sub-o--algebras of 79(/~) lies in the following fact: a Boolean sub-a-algebra ~ of 79(/~) is the range of a real projection measure F • /3(JR) --+ /2(/~), that is, ~ = F(/3(IR)) if and only if 7~ is separable [13, Lemma 3.16]. Furthermore, in that case, if ~]'~1 is a Boolean sub-a-algebra contained in ~ , then there is a real Borel function f such that ~1 = Ff(/3(R)), where F f ( x ) = F ( f - I ( x ) ) [13, Theorem 3.9], see also [2, Lemma 4.11].

Consider now two observables E1 and E defined on the o--algebras .A1 and A of the measurable spaces (f21, ,41) and (f2,.4), respectively, and taking values in /2(7-/). We say that E1 is a function of E if there is a measurable function f • f2 - + ~21 such that E1 = E f, that is, El(X) = E( f - I (X) ) for all X ~ .A1. Clearly, if E1 is a function of E, then the range of E1 is contained in the range of E. In general, for any two observables E1 and E, if E l ( . 4 ~ ) C E(A) we say that E1 is a coarse graining of E.

Assume that E1 is a coarse graining of E. If (~, E, V) is a Naimark dilation of E, we let T~ 1 be the set of all projections P e E(A) such that V*PV c El(A1). Then

Ea(A1) = V*T~IV C E(A) = V*~;(A)V.

Calling two observables equivalent if their ranges are the s ame we observe that if E(A) is a separable Boolean sub-o--algebra of 79(/~), then E is equivalent to a real projection measure F • /3(IR) --+ /2(/~). If, in addition, J-~X is a Bootean sub-o--algebra of E(A) then it can be expressed as ~-~1 = ff(/3(IR)) for some Borel function f . In this case observables E1 and E are equivalent to the two real functionally related semispectral measures E~ and E r, where E~(X)= V*Ff(X)V and Er(x) = V*F(X)V for all X E/3(IR).

The questions of interest for this study are the following. First, under what conditions is an observable separable? Secondly, if an observable is a coarse graining of another observable, when it is a function of the latter? Sections 2 and 3 are devoted to the separability questions whereas in Section 4 we study the question of functional relations between observables.

REMARK 1.1. For positive operator m e a s u r e s E1 and E, the condition EI ( ,A1) C E(-4) need not imply that E1 is a function of E. However, E1 and E may still be functionally related (functionally coexistent) so that there is a positive operator measure F with measurable functions f and g such that E1 = F o f - 1 and E = F o g-1. Indeed, as an illustration of this phenomenon, consider the real scalar measures E and E1 concentrated, respectively, on the sets {Xl, x2, x3, x4} and {Yl, y2, Y3, Y4} such that E({xl}) = E({x2}) = 1/8, E({x3}) = E({x4}) = 3/8, and EI({yl}) = EI({y2}) = EI({Y3}) = 1 /8 , EI({Y4}) = 5 / 8 . Clearly, the range of E1 is contained in that of E, but there is no function f : {Xl,X2, X3, X4} {yl, y2, y3, y4} such that El(Y) --= E(f - i (Y) ) . Indeed, if such a function exists, we must have EI({Yl}) = E(f-l({yl})) --- 1/8, which gives f - l ( {y l} ) = {xl}, or

NOTES ON COARSE GRAININGS AND FUNCTIONS OF OBSERVABLES 243

f-X({yl}) = {x2}, and EI({Y4}) = E ( f - l ( { y 4 } ) ) , which yields f - l ({y4}) = {Xl, x2, x3} or f - l ({y4}) = {Xl, x2, x4}. Both E and E1 are, however, functions of the observable {zi} ~ F({zi}) = 1/8, i = 1 . . . . . 8.

2. Separable Boolean a-algebras

In this section we collect, for the reader's convenience, some basic observations in the context of separable Boolean sub-a-algebras of the projection lattice of a Hilbert space. The proofs follow readily from known facts and the results themselves may be part of the folklore of the subject though hard to find in the literature.

Let /3 be a Boolean algebra. An atom of /3 is any nonzero element a of /3 such that b < a for b c l 3 implies b = 0 or b = a . Let At(/3) be the set of all atoms of /3. If At(/3) = 0, /3 is said to be atomless. If a and b are two different atoms of /3, then they are disjoint, a /x b = 0.

If /3i = (/3i; Oi, l i , 'i ), i = 1, 2, are Boolean a-algebras, then their Cartesian product /3 =/31 x/32 is again a Boolean a-algebra with operations defined coordi- natewise, the least and the greatest elements being 0 = (01, 02) and 1 = (11, 12), respectively.

PROPOSITION 2.1. Let /3 be a Boolean a-algebra such that every system o f mutually orthogonal nonzero elements o f / 3 is at most countable. Then /3 can be decomposed in the form /3 = 131 x/32, where /31 is a Boolean a-algebra isomorphic with the power set 2 N, where N is a finite or countable cardinal, and /32 is an atomless Boolean a-algebra.

Proof: Let At(/3) be the set of all atoms of /3. Since any two different atoms a and b of /3 are mutually orthogonal, a _< b t, 0 < IAt(/3)l ___ N0.

Define a0 := V{a " a • At(B)}; if At(B) = 0, we put a0 := 0. For any .element a • / 3 , we have the decomposition

a = (a/x a0) v (a A a~). (2.1)

Define /31 := {a • /3 • a < a0} and /32 := {a • /3 • a _ a~}. Then /31 = ! ! l a l

(/31; 0, ao, a0 ), where x 'ao := x ' / x ao for x • /31, and /32 = (/32; 0, %, 0 ), where

x'a; := x' m a~ for x • /32, are Boolean a-algebras such that /31 is isomorphic with the a-algebra 2 N, where N = [At(/3)l, and /32 is atomless. In view of (2.1) we have the decomposition /3 = / 3 1 × /32. []

The set 79(~) of all projections on ~ forms a complete orthomodular lattice with respect to the operator order and orthocomplementation P ~ P± := I~ - P, with I7~ = I and O~ = O being the identity and zero operators on ~ .

THEOREM 2.1. Let 7-[ be a complex separable Hilbert space and let /3 be a Boolean sub-a-algebra of 79(~). Then /3 is separable. In particular, i f 7-[ is finite-dimensional, then /3 = 2 N, where N is an integer such that 1 <_ N < dim 7-/.

244 A. DVURE(~ENSKIJ, R LAHTI, S. PULMANNOVA and K. YLINEN

Proof: Using Proposition 2.1 we decompose the or-algebra 13 in the form 13 = 131 x 132, where 131 is isomorphic with 2 N, N = [At(13)[, and 132 is atomless. Let P0 = V{P : P 6 At(B)} and denote ~0 = P0(7-/).

Assume dim 7-[ = N0. If P0 = Ix, then 13 = 131, and 13 is separable. If Po ~ Ix, then Ix - P0 7 ~ O, and since B2 is atomless, we have dim(7-/0) = N0. In addition, 132 is therefore a Boolean o--algebra which is a subalgebra of 79(7g~). Let 11~2 be the von Neumann algebra generated by 132. Then I~2 is a commutative von Neumann algebra acting in the infinite-dimensional complex separable Hilbert space 7-/~ and the projection lattice of I~2 coincides with 132 which is atomless. Therefore, by [12, Theorem III.1.22], ~2 is isomorphic with the von Neumann algebra L~(0, 1) (the space of all essentially bounded functions on the unit interval (0, 1) with respect to the Lebesgue measure). Since the projections from L~(0, 1) are only characteristic functions, they have a countable generator, consequently /32 has a countable generator. Because 131 is generated by the countable set of atoms, in view of 13 = B1 x 132, 13 is separable.

Assume now dim T-/< oe. Then P0 = 14 and therefore B = 131 = 2N. []

3. Separable observables

A Naimark dilation (/C, E, V) of a semisApectral measure E " .4 --+ Z2(~) is minimal if /C is the closed linear span of { E ( X ) ] X c .4}. As is well known, a minimal dilation always exists and it is unique up to an isometric isomorphism [10].

LEMMA 3.1. Let (~2, .4) be a measurable space with a separable ~r-algebra .4 and let E • .4 --~ ~ ( ~ ) be a normalized positive operator measure acting on a complex separable Hilbert space 7-[. I f (K~, E, V) is a minimal Naimark dilation of E, then ~ is separable.

Proof: Let ~" be a countable collection of subsets of f2 which generates the o--algebra .4, and let T~ be the ring generated by 5 r. Since ~- is countable, the ring T~ is countable [3, Theorem 1.5.C]. Let C be thecomplex linear span of the characteristic functions Xx of the sets X c T~, and let C be its closure in the set of bounded functions f2 -~ C (with respect to the sup-norm). C is a separable commutative C*-algebra. Let qb • C --~ ~ ( ~ ) be the positive linear map corresponding to the normalized positive operator measure E ' . 4 - ~ ~ ( ~ ) , q ~ ( f ) = f f d E . Then q~ is completely positive [10, Theorem 3.10]. Let (/C, Jr, V) be its minimal Stinespring dilation. The Hilbert space /C is separable [10, p. 46]. Let Po" 7~-~ £(K~) be the projection-valued set function defined by Po(X) = zr(Xx) for all X c 7~. Then V*Po(X)V = E(X) for all X ~ T~. From its construction it easily follows that Po is weakly g-additive on 7~.

For any ~0 c /C and X c 7¢ denote /z~,~(X)= (~olPo(X)cp). Since #~,~ is a-additive on 7-¢, it has a unique extension to a (positive) measure /z~,¢ on .4. For

1 4 "k any qg, ~ 6 /C, we let /z~0,~ ~ Y~=I : I ~q~+ik~ ,q~+ikap . Elementary estimates show

NOTES ON COARSE GRAININGS AND FUNCTIONS OF OBSERVABLES 245

that the map (q), ~p) ~ #~,~(X) is a bounded sesquilinear form for each X 6 .A, and we get a positive operator measure i " .A--+ E(/~) which extends Po.

It remains to be shown that the map i is a projection measure. We denote by M ( ~ ) the monotone class generated by 7~. The class {X E A I i ( X ) 2 = if(X)} contains T~ and is easily ~ e n to be a monotone class and so ~ equals .A [3, Theorem 1.6.B]. Clearly, V*P(X)V = E(X) for all X E .A and (/~, P, V) constitutes a minimal dilation of E and /C is separable. []

An alternative approach would be to use in the above proof Naimark's dilation theory [11, Appendix, Theorem 1] instead of Stinespring's.

REMARK 3.1. A physically relevant dilation 0E,/~, V) of a quantum observable E is typically not minimal, see e.g. [8]. An interesting example of a dilation acting on a nonseparable Hilbert space appears in [9] for the canonical phase observable.

COROLLARY 3.1. Let (f2, A) be a measurable space with a separable a-algebra A and let 7-[ be a complex separable Hilbert space. Any normalized positive operator measure E : A --~ ~ (~ ) is separable.

~Proof: Let (/C, E, V) constitute a minimal Naimark dilation of E. The set {E(X)~oIX E A, qL~ 7-/} is dense in /~. By Lemma 3.1 )U is separable. Therefore, by Theorem 2.1 E(.A) is a separable Boolean sub-o--algebra of 79(/~). []

4. Boolean observables

The Boolean structure of the range of an observable plays an important role in the functional calculus of observables. We therefore recall the following results. Here g (~ ) denotes the set of effect operators on 7g, i.e. g (7-/) = {A ~ /:(7-/) : O < _ A < I } .

PROPOSITION 4.1. The range E(J[) of an observable E : ¢ 4 - - + g,(7-[) is a Boolean subalgebra of the set g (7-[) of effects if and only if E is projection valued.

Proof: For any X 6 A the product E(X)E(X') is a positive lower bound of E(X) and E(X'). If E(A) is Boolean then E(X)/x E(X') = O, and thus E(X)E(X') = O, that is, E ( X ) 2 : E(X). On the other hand, if E is projection valued, then the claim follows from the multiplicativity of the spectral measure and from the fact that for any two projections P and R their greatest lower bound and smallest upper bound in g (7-/) are the same as in P(7-(), that is, P /x R and P v R, respectively. []

The order structure of the set of effects g (7-/) is highly complicated. For instance, if E : .A --+/2(~) is an observable, then for any X, Y 6 A, the effect E(X A Y) is a lower bound of the effects E(X) and E(Y), but these effects need not have the greatest lower bound E(X) /xc(~)E(Y) and even if E(X)AE(~)E(Y) exists it need not coincide with E(X A Y). When the order and the complement of g (~ ) are restricted to the range E(.A) of E it is possible that the system (E(.A), ___,') is a Boolean a-algebra without E being projection valued. To express that option it is

246 A. DVURECENSKIJ, E LAHTI, S. PULMANNOVA and K. YLINEN

useful to introduce two further concepts. We say that an observable E • .A -+ £(7{) is regular if for any O 4= E(X) 4= I, neither E(X) <_ E(X ' ) nor E(X' ) < E(X) , and it is A-closed if for any triple of pairwise orthogonal elements A, B, C 6 E(A), the sum A + B + C is in E(A). From [5, 1, 7] the following results are then obtained.

PROPOSITION 4.2. For any observable E • .A -+ £(7-t) the following three conditions are equivalent.

< , 1) (E(A), _, ) is a Boolean a-algebra. 2) E is regular. 3) E is A-closed.

Consider now two observables E1 and E defined on the o--algebras A1 and A of the measurable spaces ( fh , A1) and (f2, A), respectively, and taking values in £(7-/), with 7-{ being complex and separable. Assume that E1 is a coarse graining of E, that is, El(A1) C E(.A). Let (/C, E, V) be a Naimark dilation of E, with separable /C, and let 7-41 be again the set of projections P c E(A) such that V * P V c El(A1).

PROPOSITION 4.3. With the above notations, 7"41 is a Boolean sub-a-algebra of 7)(1C) if and only if there is a real Borel function f and a real semispectral measure Er such that E is equivalent with Er and E1 is equivalent with E f .

Proof: If T41 is a Boolean sub-o--algebra of P(/C) then, as a subset of E(A), it is also separable. Thus by the results [13, Lemma 3.16, Theorem 3.9] there is a real projection measure Fr and a real Borel function f such that E(A) = Fr(B(R)) and 741 = F/(13(]~)). The semispectral measures E,- := V*FrV and E f := v * F U v are now as required. The other direction is immediate. []

We say that an observable E • A --+ £(7-/) has the V-property with respect to a subset Q of E(A) if for each X, Y c .,4 and C 6 Q the inequality E(X) <_ C < E(Y) implies that there is a Z c A such that X C Z C Y and C = E(Z) . The importance of this property is in the fact that for any two (real) observables E1 and E, if El(A) C E(A) and if E has the V-property on E~(A), then E1 is a function of E [6].

LEMMA 4.1. With the above notation, Olc, Itc E 74~, and if P ~ 741 then also P± E T4i. Moreover, forNany P, R E 741, if P < R, then V * P V < V*RV. In addition, the observable E has the V-propety on 741.

Proof: If P ~ 741, then V * P V = E l (X) for some X ~ .A~ and thus

El(X' ) --- I~ - El(X) = V * V - V * P V = V*(I/c - P)V,

so that P± 6 741. If P _< R, then for any q 6/C, (7t I P7 t ) < ( q [ R7 r), and thus, in particular, for any ~0 6 7-/,

< I E l ( x m > = I v*P = < I P

<_ < V~ I RV~ } = ( ~ I V*RV~ : {~oI E~(Y)~o).

NOTES ON COARSE GRAININGS AND FUNCTIONS OF OBSERVABLES 247

To demonstrate the V-property, let X, Y %-4, X _ Y, so that fiT(X) < ffT(Y~. Assume that P 6 ~ l is such that E(X) < P < E(Y) . Let Z ~ .4 be such that E(Z) = P. Then for Z I = X U ( Y A Z ) we have X _ Z I _ Y , and

E(Z1) = E(X) v (E(Y) A E(Z)) = (E(X) v E(Y)) A (E(X) v P) = E(Y) A P = P. []

REMARK 4.1. The assumption that E has the V-property on T£] does not imply that E has the V-property on El(.4). For an illustration, see Remark 1.1.

PROPOSITION 4.4. With the @ove notation, if E1 is projection valued, then ~r~ 1 is a Boolean sub-a-algebra of E(.4).

Proof: For any P 6 79(/C), V * P V c 7:'(7-t) if and only if V V * P = PVV* . Let P , R 6 ~1 so that there are X,Y 6 .41 such that V * P V = E l (X) and V*RV = El(Y) . Then

V*P /x R V = V * P R V = V * V V * P R V

= V * P V V * R V = E] (X)EI (Y )

= E1 (X r3 Y)

showing that ~1 is closed under A. By the de Morgan laws, the same is true for v. If (Pn)~l is a sequence of mutually orthogonal projections of 7~1, that is, Pn < Pm z for all n 7~m, then also EI (X~)< EI(Xm) J - = EI(X~n). Therefore,

v*(VPn)V = V*(ZPn)V = ~ V*PaV = ZEI(Xn)= EI(UXn ) (where the series converge weakly) which shows the a-property of "/~1. []

COROLLARY 4.1. Let f21 and f2 be complete separable metric spaces and let B(f21) and 13(f2) be their respective Borel a-algebras. Assume that ~-~1 and f2 have the cardinality of R. Consider the observables E1 " ~(~1) --+ L;(7-/) and E ' B ( f 2 ) - + / ; (~ ) such that E1 is a coarse graining of E. I f E1 is projection valued, then E1 = E f for some Borel function f • ~2 ~ f21.

Proof: Since f21 and f2 are complete separable metric spaces with the cardinality of N, according to [4, Remark (ii), p. 451], there are bijections ot • f2 --+ N and fl • ~"21 ~ • which are such that ~, o~ -1, r , and fi-1 are Borel measurable. Now E" and E~ are real observables with the same ranges as E1 and E, respectively. By [13, Theorem 3.9] there is a measurable function g : R -+ R such that E~(X) = E~(g- I (X) ) , X c /3(N). Putting X = f l(Z), Z c 13(f2), we obtain El (Z) = E~(f l (Z)) = E~(g- l ( f i (Z ) ) ) = E Z ( z ) , where f = fl-1 ogoot "f2 -+ f21. []

Acknowledgement The authors are grateful to Dr. J. Hamhalter, Technical University of Prague, for

useful discussions on von Neumann algebras. The paper has partially been supported by the grant VEGA No 2/3163/23, Slovak Academy of Sciences, and by the Science Technology Assistance Agency under contract No APVT-51-032002, Bratislava.

248 A. DVURECENSKIJ, P. LAHTI, S. PULMANNOV,~ and K. YLINEN

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