On the infimum of quantum effects

On the infimum of quantum effects

Aurelian Gheondea^a兲

Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey and Institutul de Matematică al Academiei Române, C.P. 1-764, 014700 București, România Stanley Gudder^b兲

Department of Mathematics, University of Denver, Denver, Colorado 80208 Peter Jonas^c兲

Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany

共Received 24 September 2004; accepted 21 March 2005; published online 12 May 2005兲 The quantum effects for a physical system can be described by the setE共H兲 of positive operators on a complex Hilbert space H that are bounded above by the identity operator. While a general effect may be unsharp, the collection of sharp effects is described by the set of orthogonal projections P共H兲債E共H兲. Under the natural order, E共H兲 becomes a partially ordered set that is not a lattice if dim H 艌2. A physically significant and useful characterization of the pairs A,B苸E共H兲 such that the infimum A∧B exists is called the infimum problem. We show that A∧P exists for all A苸E共H兲, P苸P共H兲 and give an explicit expression for A∧P.

We also give a characterization of when A∧共I−A兲 exists in terms of the location of the spectrum of A. We present a counterexample which shows that a recent con- jecture concerning the infimum problem is false. Finally, we compare our results with the work of Ando on the infimum problem. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1904704兴

I. INTRODUCTION

A quantum mechanical measurement with just two values 1 and 0共or yes and no兲 is called a quantum effect. These elementary measurements play an important role in the foundations of quantum mechanics and quantum measurement theory.3–5,7,14,16,18

We shall follow the Hilbert space model for quantum mechanics in which effects are represented by positive operators on a complex Hilbert spaceH that are bounded above by the identity operator I. In this way the set of effectsE共H兲 becomes

E共H兲 = 兵A 苸 B共H兲:0 艋 A 艋 I其.

The set of orthogonal projections P共H兲債E共H兲 corresponds to sharp effects while a general A 苸E共H兲 may be unsharp 共fuzzy, imprecise兲. Employing the usual order A艋B for the set of bounded self-adjoint operatorsS共H兲 on H, we see that 共E共H兲, 艋兲 is a partially ordered set. It is well known that共E共H兲, 艋兲 is not a lattice if dim H艌2. However, if the infimum A∧B of A , B 苸E共H兲 exists then A∧B has the important property of being the largest effect that physically implies both A and B. It would thus be of interest to give a physically significant and useful characterization of when A∧B exists. This so-called infimum problem has been considered for at least 10 years.2,10–12,17,19

Before discussing the progress that has been made toward solving the infimum problem, let us compare the situation with that of the partially ordered set 共S共H兲, 艋兲. Of course, if A,B

a兲Electronic mail: aurelian@fen.bilkent.edu.tr and gheondea@imar.ro

b兲Electronic mail: sgudder@math.du.edu

c兲Electronic mail: jonas@math.tu-berlin.de

46, 062102-1

0022-2488/2005/46共6兲/062102/11/$22.50 © 2005 American Institute of Physics

苸S共H兲 are comparable, that is, A艋B or B艋A, then A∧B exists and is the smaller of the two. A surprising result of Kadison¹⁵states that the converse holds. Thus, for A , B苸S共H兲, A∧B exists in S共H兲 if and only if A and B are comparable. We conclude that 共S共H兲, 艋兲 is an antilattice which is as far from being a lattice as possible. The situation is quite different in共E共H兲, 艋兲. In fact it is well known that P∧Q exists inE共H兲 for all P,Q苸P共H兲. More generally, we shall show that A∧P exists inE共H兲 for all A苸E共H兲, P苸P共H兲 and give an explicit expression for A∧P. The existence of A∧P has already been proved in Ref. 18 but we present a different proof here.

For A , B苸E共H兲 let PA,B be the orthogonal projection onto the closure of Ran共A^1/2兲艚Ran共B^1/2兲. It is shown in Ref. 19 that if dim H⬍⬁ then A∧B exists in E共H兲 if and only if A∧P_A,Band B∧P_A,Bare comparable and in this case A∧B is the smaller of the two. This was considered to be a solution to the infimum problem for the case dimH⬍⬁ and it was conjectured in Ref. 19 that this result also holds in general. One of our main results is that this conjecture is false. We shall present an example of a pair A , B苸E共H兲 with dim H=⬁ for which A∧B exists in E共H兲 but A∧P_A,B and B∧P_A,B are not comparable. In addition, we prove that, assuming A∧B exists, P_A,Bis the smallest of all orthogonal projections P having the property that 共A∧P兲∧共B∧P兲 exists and 共A∧P兲∧共B∧P兲=A∧B. Combined with the counter-example as de- scribed before, this means that, in the infinite dimensional case, there is no orthogonal projection to replace P_A,Band have a positive solution to the infimum problem.

The negation A⬘of an effect A is defined to be the effect A⬘= I − A. Physically, A⬘is the effect A with its values 1 and 0 reversed. We also present a simple spectral characterization of when A∧A⬘^{exists in}E共H兲. The result is essentially the same as Theorem 2 in Ref. 2, with the difference that we express the condition in terms of the location of the spectrum of A and the proof is based on the matrix representations obtained in the preceding section.

Ando has given a solution to the infimum problem in terms of a generalized shorted operator.² However, in our opinion, these shorted operators do not have a physical significance in contrast to the operationally defined operators A∧P_A,B and B∧P_A,B. Finally, we discuss the relationship between our work and that of Ando. First, we show that the shorted operator of A by B is always smaller than A∧P_A,B. Actually, it is the fact, that in the infinite dimensional case, the shorted operator of A by B can be strictly smaller than A∧P_A,B, that is responsible for the failure of a solution of the infimum problem similar to the finite dimensional case. This can be viewed from the counter-example as before, but we record also a simpler one that illustrates this situation.

We now briefly discuss connections between the infimum problem and physics. Quantum effects have been studied by mathematicians and physicists for over 40 years.^5,16,17 Besides the applications of effect-valued measures in quantum measurement theory, many researchers consider effects to be the basic elements of important quantum structures. In recent times quantum effects have been organized into a structure called an effect algebra^7,10 and their order properties have been studied.^11,12,17Among other things, the effect algebra E共H兲 is a partially ordered set and if A∧B exists for A , B ,苸E共H兲, then this effect has important physical properties. In particular, among all the effects that have a smaller probability of occurring than both A and B, A∧B has the largest probability. Thus if A∧B exists, then A∧B has a crucial physical significance. In the case where A and B are sharp, A and B are projections, A∧B always exists and is the projection onto the intersection of their ranges. But if A and B are not sharp, the situation is much more compli- cated. An interesting special case is when A苸E共H兲 and P苸P共H兲. In this case A∧P always exists and if A and P commute共are compatible兲 then A∧P = AP. However, if A and P do not commute an explicit closed form expression for A∧P has been difficult to obtain and is now presented in Theorem 2.2. We can now define conditional probabilities

prob共A兩P兲 = prob共A∧P兲/prob共P兲

and conditional measurements and these may have useful physical applications. Finally, our Ex- ample 4.2 gives a surprising phenomenon that does not occur in finite dimensional Hilbert spaces.

The existence of effects such as those in this example may have interesting physical significance.

II. INFIMUM OF A QUANTUM EFFECT AND A SHARP EFFECT

We first record a parametrization of bounded positive 2⫻2 matrices with operator entries, in terms of operator balls.

In the following we make use of the Frobenius-Schur factorization: for T, X, Y, Z bounded operators on appropriate spaces and T boundedly invertible, we have

冋

册

冋

册冋

册冋

册

For instance, by using Frobenius–Schur factorizations and a perturbation argument one can obtain the following classical result of Shmulyan.²¹

Theorem 2.1: Let A苸B共H兲 be self-adjoint and H=H1丣H2an orthogonal decomposition of H. Then A艌0 if and only if it has a matrix representation of the following form:

A =

冋

册

In addition, the operator ⌫ can be chosen in such a way that Ker共⌫兲傶Ker共A2兲 and Ker共⌫^*兲傶Ker共A1兲, and in this case it is unique.

For two effects A , B苸E共H兲 we denote by A∧B, the infimum, equivalently, the greatest lower bound, of A and B over the partially ordered set共E共H兲, 艋兲, if it exists. To be more precise, A∧B is an operator inE共H兲 uniquely determined by the following properties: A∧B艋A, A∧B艋B, and an arbitrary operator D苸E共H兲 satisfies both D艋A and D艋B if and only if D艋A∧B. Charac- terizations of the existence of infimum for positive operators have been obtained for the finite- dimensional case in Ref. 19, and in general in Ref. 2.

In Theorem 4.4 of Ref. 19 it is proved that the infimum A∧P exists for any A苸E共H兲 and P苸P共H兲. As a consequence of Theorem 2.1 we can obtain an explicit description of A∧P, together with another proof of the existence.

Theorem 2.2: For any A苸E共H兲 and P苸P共H兲 the infimum A∧P exists, more precisely, if A has the matrix representation as in 共2.2兲 with respect to the orthogonal decomposition H

= Ran共P兲丣Ker共P兲, where A1苸E共Ran共P兲兲, A2苸E共Ker共P兲兲, and ⌫苸B共Ker共P兲, Ran共P兲兲, with

储⌫^储艋1, Ker共⌫兲傶Ker共A2兲 and Ker共⌫^*兲傶Ker共A1兲, then

A∧P =

冋

册

苸B共Ran共P兲兲⁺, A₂苸B共Ker共P兲兲⁺, and⌫苸B共Ker共P兲,Ran共P兲兲, with^储⌫^储艋1, Ker共⌫兲傶Ker共A2兲 and Ker共⌫^*兲傶Ker共A1兲. Since A艋I it follows that A1艋IRan共P兲and A₂艋IKer共P兲. Consider the operator D苸B共H兲, defined by the matrix in 共2.3兲. Clearly 0艋D艋 P, in particular D苸E共H兲. In addition,

A − D =

冋

1/2⌫^*A₁^1/2 A₂

册

hence A艌D.

Let C苸E共H兲 be such that C艋A, P. From C艋 P it follows that CP= PC=C and hence

C =

冋

册

Then

0艋 A − C =

冋

册

The matrix with operator entries in共2.4兲 can be factored as

冋

册 冋

册 冋

册

Note that by Ker共⌫兲傶Ker共A2兲 or, equivalently, Ran共⌫^*兲債Ran共A2兲, A−C and each of the factors of共2.5兲 map the subspace H⬘^{= Ran共P兲丣Ran共A}2兲 into itself. Since diag共IRan共P兲A₂^1/2兲 regarded as an operator onH⬘, is symmetric and has dense range, A − C艌0 implies that the middle term in 共2.5兲 regarded as an operator inH⬘ is non-negative. By performing a Frobenius–Schur factorization of this middle term, we find A₁^1/2⌫⌫^*A₁^1/2艋A1− C₁, that is, C₁艋A1

1/2共IRan共P兲−⌫⌫^*兲A1

1/2, or, equiva- lently, C艋D.

We thus proved that A∧P exists and has the matrix representation as in共2.3兲. 䊐 Remark 2.3: If A苸E共H兲, EA is the spectral function of A and⌬ is a Borel subset of 关0, 1兴, then A∧E_A共⌬兲=AEA共⌬兲. This is an immediate consequence of Theorem 2.2. The second to last sentence in the proof of Theorem 2.2 can also be demonstrated by using the well-known fact that any operator matrix of the form

冋

册

is positive if and only if A艌0 and BB^*艋A.

Let A , B苸E共H兲. By PA,B we denote the orthogonal projection onto the closure of Ran共A^1/2兲艚Ran共B^1/2兲. As mentioned in the introduction, the infimum problem for a finite dimen- sional space H was solved in Ref. 19 by showing that A∧B exists if and only if A∧P_A,B and B∧P_A,B are comparable, and that A∧B is the smaller of A∧P_A,B and B∧P_A,B. The following proposition shows that for dimH=⬁ the infimum problem for A and B can be reduced to the same problem for the “smaller” operators A∧P_A,Band B∧P_A,B. In Sec. IV we will see that in this case the infimum problem cannot be solved in the same fashion, as conjectured in Ref. 19.

Proposition 2.4: Let A , B苸E共H兲. Then A∧B exists if and only if共A∧P_A,B兲∧共B∧P_A,B兲 exists.

In this case A∧B =共A∧P_A,B兲∧共B∧P_A,B兲.

Proof: Note first that the operators A∧P_A,Band B∧P_A,Bexist, e.g., by Theorem 2.2.

Let us assume that 共A∧P_A,B兲∧共B∧P_A,B兲 exists and let C苸E共H兲 be such that C艋A,B, thus we have Ran共C^1/2兲債Ran共A^1/2兲艚Ran共B^1/2兲債Ran共PA,B兲 and hence C艋 PA,B. Therefore, C 艋A∧P_A,B and C艋B∧P_A,B and hence, by the majorization theorem as in Ref. 6, C 艋共A∧P_A,B兲∧共B∧P_A,B兲. Taking into account that 共A∧P_A,B兲∧共B∧P_A,B兲艋A,B it follows that A∧B exists and equals共A∧P_A,B兲∧共B∧P_A,B兲.

Conversely, let us assume that A∧B exist. Then, A∧B艋 PA,B. This relation and A∧B艋A,B give A∧B艋A∧P_A,B, A∧B艋B∧P_A,B. Let C苸E共H兲 be such that C艋A∧P_A,B, B∧P_A,B. Then C

艋A, B, PA,Band, in particular, C艋A∧B. 䊐

One may ask for which orthogonal projections P except P_A,Bthe statement of Proposition 2.4 is true. It turns out that P_A,Bis the infimum of the set of those projections P.

Theorem 2.5: Let A , B苸E共H兲 such that A∧B exists. Let ⌸A,Bbe the set of all orthogonal projections subject to the properties that共A∧P兲∧共B∧P兲 exists and 共A∧P兲∧共B∧P兲=A∧B. Then

⌸A,B=兵P 苸 P共H兲兩PA,B艋 P其.

In order to prove the above stated proposition, we first consider the connection of parallel sum with the infimum of quantum effects共see also Ref. 2兲. To see this, instead of giving the original definition as in Ref. 8, we prefer to introduce the parallel sum of two quantum effects by means of the characterization of Pekarev–Shmulyan,²⁰

具共A:B兲h,h典 = inf兵具Aa,a典 + 具Bb,b典兩h = a + b其,for all h 苸 H. 共2.7兲 Theorem 2.6:共Refs. 8 and 20兲 Let A,B苸B共H兲⁺. Then

共i兲 0艋A: B艋A, B, 共ii兲 A : B = B : A,

共iii兲 Ran共共A:B兲^1/2兲=Ran共A^1/2兲艚Ran共B^1/2兲,

共iv兲 if A₁, B₁苸B共H兲⁺are such that A艋A1 and B艋B1, then A : B艋A1: B₁, 共v兲 if A + B is boundedly invertible, then储A:B储=A共A+B兲⁻¹B,

共vi兲 If A_n&A and Bn&B strongly, then An: B_n&A:B strongly.

In view of the properties of the parallel sum listed above, a moment of thought shows that if P , Q苸P共H兲, that is, P and Q are orthogonal projections in H, then P∧Q overE共H兲 always exists and coincides with the orthogonal projection onto Ran共P兲艚Ran共Q兲. By Theorem 4.3 in Ref. 8 we also have P∧Q = 2共P:Q兲.

Lemma 2.7: Let A , B苸E共H兲 be such that A∧B exists. Then 共i兲 Ran共共A∧B兲^1/2兲=Ran共共A:B兲^1/2兲,

共ii兲 共A∧B兲^1/2=共A:B兲^1/2V for some boundedly invertible operator V苸B共H兲, 共iii兲 A : B艋A∧B艋␥共A:B兲, for some␥⬎0.

Proof: Since A∧B艋A it follows that Ran共共A∧B兲^1/2兲債Ran共A^1/2兲. Similarly we have Ran共共A∧B兲^1/2兲債Ran共B^1/2兲, hence Ran共共A∧B兲^1/2兲債Ran共A^1/2兲艚Ran共B^1/2兲=Ran共共A:B兲^1/2兲.

For the converse inclusion, note that A : B艋A and A:B艋B; since A:B艋A:I=A共A+I兲⁻¹艋A.

Thus, by the definition of A∧B, it follows that A : B艋A∧B. In particular, this proves that Ran共共A∧B兲^1/2兲傶Ran共共A:B兲^1/2兲, and hence 共i兲 is proved.

The assertions共ii兲 and 共iii兲 are consequences of 共i兲 and the majorization theorem as in Ref.

6. 䊐

Lemma 2.8: If A , B苸E共H兲 and A∧B exists, then A∧B艋 PA,B and Ran共A∧B兲 is dense in Ran共PA,B兲.

Proof: This is a consequence of Theorem 2.6 and Lemma 2.7. 䊐 We now come back to Theorem 2.5.

Proof of Theorem 2.5: Let P苸⌸A,B. Then A∧B艋 P and hence Ran共A∧B兲債Ran共P兲. There- fore, by Lemma 2.8 Ran共PA,B兲債Ran共P兲, that is, PA,B艋 P.

Assume that P艌 PA,B. We claim that then 共A∧P兲∧共B∧P兲 exists and it coincides with 共A∧P_A,B兲∧共B∧P_A,B兲. Evidently, 共A∧P_A,B兲∧共B∧P_A,B兲艋A∧P , B∧P. Let C苸E共H兲 with C 艋A∧P, B∧P. Then C艋A∧B艋 PA,Band hence,

C艋 共A∧P_A,B兲∧共B∧P_A,B兲.

Therefore,共A∧P兲∧共B∧P兲 exists and, by Proposition 2.4 it coincides with A∧B. 䊐 III. INFIMUM OF A QUANTUM EFFECT AND ITS NEGATION

The negation A⬘of an effect A is defined to be the effect A⬘= I − A. Physically, A⬘is the effect A with its values 1 and 0 reversed. In the following we present a characterization of when A∧A⬘ exists in E共H兲 in terms of the location of the spectrum of A. The theorem essentially coincides with the result of Ando 共Ref. 2, Theorem 2兲, the difference consists on that we express the condition with the help of the spectrum of A and the proof is based on the matrix representations as in Sec. II. There is also a similar characterization in Ref. 13.

Theorem 3.1: Let A be a quantum effect on the Hilbert spaceH. Then the following asser- tions are equivalent:

共i兲 A∧共I−A兲 exists,

共ii兲 ␴共A兲, the spectrum of A, is contained either in 兵0其艛关₂^{1, 1}兴^{or in}关^{0 ,1}₂兴艛兵1其,

共iii兲 A∧P_A,I−A and 共I−A兲∧P_A,I−Aare comparable, that is, either A∧P_A,I−A艋共I−A兲∧P_A,I−Aor 共I−A兲∧P_A,I−A艋A∧P_A,I−A.

In addition, if either of the above holds, letting g苸C共关0,1兴兲 be the function

g共t兲 = min共t,1 − t兲 =

再

冎

we have, by continuous functional calculus, A∧共I−A兲=g共A兲.

Proof: Let E_A denote the spectral function of A. In view of Proposition 2.4, A∧共I−A兲 exists if and only if 共A∧P_A,I−A兲∧共共I−A兲∧P_A,I−A兲 exists. A moment of thought shows that PA,I−A

= E_A共共0,1兲兲 and hence, by Remark 2.3, we have that A∧P_A,I−A= AE_A共共0,1兲兲 and 共I−A兲∧P_A,I−A

=共I−A兲EA共共0,1兲兲. Thus, without restricting the generality, we can and will assume in the follow- ing that 0 and 1 are not eigenvalues of A. Now, the equivalence of共ii兲 with 共iii兲 is a matter of elementary spectral theory for selfadjoint operators, hence we will prove only the equivalence of 共i兲 and 共ii兲.

To prove that共ii兲 implies 共i兲, let us assume that␴共A兲 is contained either in 兵0其艛关₂¹, 1兴or in 关0 ,¹₂兴艛兵1其. To make a choice, let us assume that␴共A兲債兵0其艛关¹₂, 1兴. Since, by assumption, 0 is not an eigenvalue of A, it follows that ␴共A兲債关¹₂, 1兴. Then A艌I−A and clearly A∧共I−A兲=I−A

= g共A兲, where the function g is defined as in 共3.1兲. A similar argument holds in case we assume

␴共A兲債关0 ,¹₂兴艛兵1其; in this case A∧共I−A兲=A=g共A兲.

Conversely, let us assume that A∧共I−A兲=D, the infimum of A and I−A over E共H兲, exists.

Using the spectral measure E_A of A, let E₁= E_A共关0,1/2兴, A1= A兩E1H, E2= E_A共共1/2,1兴兲, A2

= A兩E2H. We write D as an operator matrix with respect to the decomposition H=E1H丣E₂H,

D =

冋

册

with contractive⌫苸B共E2H,E1H兲, cf. Theorem 2.1. Since g共A兲艋A, I−A, by the definition of D we have

0艋 D − g共A兲 =

冋

1/2⌫^*D₁^1/2 D₂−共I2− A₂兲

册

Therefore, 0艋D1− A₁ while taking into account that D艋A it follows that D1艋A1, hence D₁

= A₁. Similarly, 0艋D2−共I2− A₂兲 and, since D艋I−A it follows D2艋I2− A₂, hence D₂= I₂− A₂. Thus, the main diagonal of the matrix in共3.2兲 is null, hence 共e.g., by Theorem 2.1兲 it follows that D = g共A兲.

Further, let␧苸共0,1/4兲, and consider the operators

E_␧,1= E_A共共␧,− ␧ + 1/2兲兲, E_␧,2= E_A共共␧ + 1/2,1 − ␧兲兲. 共3.3兲 Denote E_␧= E_␧,1+ E_␧,2and A_␧= A兩E_␧H. We show that A_␧∧共I−A_␧兲 exists. To see this, we remark that, as proven before, g共A兲=A∧共I−A兲, so we actually show that D_␧= D兩E_␧H=g共A_␧兲 coincides with A_␧∧共I−A_␧兲. Indeed, assume that for some C_␧苸E共E_␧H兲 we have C_␧艋A_␧, I − A_␧. Then, letting C = C_␧E_␧苸E共H兲 it follows that C艋A, I−A. Since D=A∧共I−A兲 this implies C艋D and hence C_␧艋D_␧. Therefore, D_␧coincides with A_␧∧共I−A_␧兲.

We finally prove that 共i兲 implies 共ii兲. Assume that 共i兲 holds and 共ii兲 is not true. Then there exists␧苸共0,1/4兲 such that E_␧,1⫽0 and E_␧,2⫽0, where we use the notation as in 共3.3兲. Letting

A_␧,1= A兩E_␧,1H, A_␧,2= A兩E_␧,2H,

and d =␧共1+

冑

C =

冋

^冑

^冑

册

冋

^冑

^冑

^冑

冑

册

=

冋

册

^冑

冋

册

we have

A_␧− C =

冋

^冑

^冑

册

^冋

^册

冋

^冑

^冑

册

and

I − A_␧− C =

冋

^冑

^冑

册

^冋

^册

冋

^冑

^冑

册

艌 0.

But, the operator

共A_␧∧共I_␧− A_␧兲兲 − C = g共A_␧兲 − C = d

冋

^冑

^冑

册

is not non-negative for some choices of T, unless at least one of the spectral projections E_␧,1and E_␧,2 is trivial. Since␧ is arbitrarily small, it follows that A cannot simultaneously have spectral

points in共0,1/2兲 and 共1/2,1兲. Therefore, 共i兲 implies 共ii兲. 䊐

IV. TWO EXAMPLES

In this section we answer in the negative a question raised in Ref. 19. First we recall how the problem of the existence of the infimum of A and B in E共H兲 can be reduced to the infimum problem for some quantum effects and their negations. Assume, in addition, that Ker共A+B兲=0.

Let f_A+Bbe the affine 共that is, linear on convex combinations兲 mapping defined as in Ref. 9 by f_A+B:兵C兩0 艋 C 艋 A + B其 → 兵D兩0 艋 D 艋 PA+B其, 共4.1兲 with C =共A+B兲^1/2f_A+B共C兲共A+B兲^1/2. By Theorem 2.2 in Ref. 9, f_A+Bis well defined. Since f_A+B is an affine isomorphism, A∧B exists if and only if f_A+B共A兲∧f_A+B共B兲 exists. As

f_A+B共A兲 + fA+B共B兲 = fA+B共A + B兲 = I we are in the situation of Theorem 3.1.

Actually, the following more general fact holds.

Lemma 4.1: Let A苸E共H兲, 0艋C, D艋A, and consider the mapping fA as defined in 共4.1兲.

Then C∧D exists if and only if f_A共C兲∧f_A共D兲 exists and, in this case, we have

f_A共C∧D兲 = fA共C兲∧f_A共D兲.

Proof: This is a consequence of Theorem 2.5 in Ref. 9. 䊐

By Proposition 2.4, the infimum of A and B exists if and only if the infimum of A∧P_A,Band B∧P_A,B exists or, equivalently, the infimum of the restrictions A˜ ªA∧P_A,B兩 PA,BH and B˜

ªB∧P_A,B兩 PA,BH exists. Since Ker共A˜+B˜兲=兵0其, A˜∧B˜ exists if and only if f_{A˜ +B˜}共A˜兲∧f_{A˜ +B˜}共B˜兲 exists, and for the pair f_{A˜ +B˜}共A˜兲, fA˜ +B˜共B˜兲 we observe that Theorem 3.1 applies. Therefore, under the additional assumptions that 0 and 1 are not eigenvalues of f_{A˜ +B˜}共A˜兲 and fA˜ +B˜共B˜兲, A˜∧B˜ exists if and only if A∧P_A,Band B∧P_A,Bare comparable; in this case, A∧B coincides with the smaller of the

A∧P_A,B and B∧P_A,B. For a finite dimensional Hilbert space it was proven in Ref. 19 that the infimum of the operators A∧P_A,Band B∧P_A,Bexists if and only if they are comparable.

The next example shows that, contrary to the finite dimensional case, we may have two quantum effects B₁ and B₂ for which B₁∧B₂ exists, but 共B1∧P_B

1,B₂兲 and 共B2∧P_B

1,B₂兲 are not comparable.

Example 4.2: LetH=L²关−1,1兴 and A be the operator of multiplication with the square of the independent variable on H, 共Ax兲共t兲=t²x共t兲, for all x苸L²关−1,1兴. Then A is a non-negative con- traction onH, that is, a quantum effect, and the same is its square root A^1/2, that is,共A^1/2x兲共t兲

=兩t兩x共t兲, x苸L²关−1,1兴. Note that A, and hence A^1/2, are injective.

Let 1 be the constant function equal to 1 on 关−1,1兴, ␪共t兲ªsgn共t兲, and ␹±ª2¹共1±␪兲, the characteristic functions of关0, 1兴 and, respectively, 关−1,0兴. All these functions are in L²关−1,1兴.

Note that 1 and␪ span the same two dimensional space as␹±. Denote H0=H両span兵1,␪其 = H両span兵␹+,␹−其.

With respect to the decomposition

H = C1丣C␪丣H0

consider two quantum effects C₁ and C₂onH defined by

C₁=

冤

冥

冤

冥

where I₀is the identity operator onH0. Clearly we have C₁+ C₂= I and letting B₁= A^1/2C₁A^1/2, B₂= A^1/2C₂A^1/2,

we have

B₁+ B₂= A.

Comparing the spectra of C₁and C₂and using Theorem 3.1, it follows that C₁∧C₂exists, but C₁and C₂are not comparable. Therefore, using Lemma 4.1, it follows that B₁∧B₂exists, but B₁ and B₂ are not comparable. In the following we will prove that P_B

1,B₂= I, that is, Ran共B1

1/2兲艚Ran共B2

1/2兲 is dense in H. We divide the proof in several steps.

Step 1: A^1/2H0is dense inH.

Indeed, let f苸H=L²关−1,1兴 be a function such that for all h0苸H0we have 0 =具A^1/2h₀, f典 = 具h0,A^1/2f典.

Then A^1/2f is a linear combination of the functions 1 and␪, that is, there exist scalars␣^and␤^such that

兩t兩f共t兲 =␣⁺␤^sgn共t兲, t 苸 关− 1,1兴 and hence

f共t兲 =␣⁺␤^sgn共t兲

兩t兩 =

冦

冧

Since f苸L²关−1,1兴 this shows that f =0 and the claim is proven.

Let us consider the following linear manifolds inH:

Fª 兵f 苸 L²关− 1,1兴兩f piecewise constant其,

F0ª 兵f 苸 F兩∃ ␧ ⬎ 0 s . t . f兩共− ␧,␧兲 = 0,具f,␹−典 = 具f,␹+典 = 0其.

Step 2: F0is dense inH0.

Indeed, to see this, let us first note that F0傺H0. If h₀ is an arbitrary vector in H0 and ␧

⬎0, there exists f1苸F such that

储h₀− f_{1 储}艋␧

8 hence兩具h0− f₁,␹±典兩 艋␧

8. 共4.2兲

Moreover, there exists f₂苸F such that it is zero in a neighbourhood of zero and

储f₁− f_{2 储}艋 ␧

8. 共4.3兲

Consequently,

储h₀− f_{2 储}艋␧

4 and hence兩具h0− f₂,␹±典兩 艋␧

4. 共4.4兲

Let

f₃= f₂+ 2␹_关1/2,1兴具h0− f₂,␹+典 + 2␹_{关−1,−1/2兴}具h0− f₂,␹−典.

Then, from the choice of f₂ it follows

具f3,␹+典 = 具f2,␹+典 + 具h0− f₂,␹+典 = 具h0,␹+典 = 0 and

具f3,␹−典 = 具f2,␹−典 + 具h0− f₂,␹−典 = 具h0,␹−典 = 0, hence f₃苸F0. Finally, from共4.2兲, 共4.3兲, and 共4.4兲 we get

储h₀− f_{3 储}艋^储h₀− f_{1 储}+储f₁− f_{2 储}+储f₂− f_{3 储}艋 ␧, and the claim is proven.

Finally, we prove the following.

Step 3: P_B

1,B₂= I, that is, Ran共B1

1/2兲艚Ran共B2

1/2兲 is dense in H.

In the following we are using the inverse operator A^−1/2on its range. By the preceding claim, A^1/2共A^−1/2F0兲 is a linear submanifold in H0and dense in it. Since the restrictions of C₁and C₂to H0coincide with ¹₂I₀, it follows that the linear manifolds C₁A^1/2共A⁻¹F0兲 and C2A^1/2共A⁻¹F0兲 coin- cide and are dense in H0. Consequently, the linear manifolds A^1/2C₁A^1/2共A⁻¹F0兲 and A^1/2C₂A^1/2共A⁻¹F0兲 coincide and, by Step 1 and Step 2, they are dense in H. Thus, the linear manifold,

L = B1共A^−1/2F0兲 = B2共A^−1/2F0兲 債 Ran共B1兲 艚 Ran共B2兲 債 Ran共B1

1/2兲 艚 Ran共B2 1/2兲, is dense inH. This concludes the proof of the last step, and the example.

In order to explain the connection with the characterization of the existence of infimum obtained by Ando in Ref. 2 we consider the comparison of A∧P_A,Bwith the generalized shorted operator, as considered in Ref. 2.

Lemma 4.3: Let A , B苸E共H兲. Then, for any sequence ␣n of positive numbers that converge increasingly to infinity, we have

SO- lim

n→⬁共A:␣nB兲 艋 A∧P_A,B, 共4.5兲

and the limit does not depend on the sequence共␣n兲.

Proof: First note that the sequence of positive operators A :␣nB is nondecreasing and bounded by A, cf. Ref. 8. Consequently, the strong operator limit exists and does not depend on the sequence ␣n increasing to infinity. We thus can take ␣n= n. Since the parallel sum is strongly continuous in the second variable with respect to nondecreasing sequences, cf. Theorem 2.6, we have A : nB艋A and, since Ran共共A:nB兲^1/2兲=Ran共A^1/2兲艚Ran共B^1/2兲 it follows A:nB艋 PA,B and

hence共4.5兲 holds. 䊐

Given two positive operators A and B, the generalized shorted operator关B兴A is defined 共see Ref. 1兲 by

关B兴A = lim

n→⬁A:共nB兲.

The main result in Ref. 2 states that the infimum A∧B exists if and only 关B兴A and 关A兴B are comparable and, in this case, A∧B is the smaller of关A兴B and 关B兴A. In view of this result and our Example 4.2, it follows that, in general, 共4.5兲 cannot be improved to equality. Here we have a simpler example emphasizing this fact.

Example 4.4: LetH=L²关0,1兴 and A the operator of multiplication with the function t². Then A is bounded, contractive, and positive. In addition, A^1/2is the operator of multiplication with the independent variable t. Note that both A and A^1/2are injective.

Further, let 1 be the function constant 1 in L²关0,1兴 and note that it does not belong to the range of either A or A^1/2. Let C be a non-negative contraction in H with kernel C1 and define B = A^1/2CA^1/2. Then the operator B is injective and hence its range is dense in H. Since Ran共B兲債Ran共B^1/2兲 and, by construction, Ran共B兲債Ran共A^1/2兲 as well, it follows that Ran共A^1/2兲艚Ran共B^1/2兲 is dense in H, hence PA,B= I.

For each n艌1 consider the function vn苸L²关0,1兴 defined by

vn共t兲 =

再

冎

Note that A^1/2v_n=␹_共1/n,1兴, the characteristic function of the interval 共1/n,t兴. Taking into account that the sequence of functions␹共1/n,1兴 converges in norm to the function 1, it follows that

具Bvn,v_n典 = 具CA^1/2v_n,A^1/2v_n典 = 具C␹_共1/n,1兴^,␹_共1/n,1兴典 → 具C1,1典 = 0.

Let ␣n be a sequence of positive numbers increasing to +⬁ and such that ␣n具Bvn, Bvn典 converges to 0. It is easy to see that this is always possible. Then using the characterization of the parallel sum as in Theorem 2.6.共vi兲, for arbitrary n艌m⬎2 we have

具共A:␣nB兲vmvm典 = inf兵具Au,u典 +␣n具Bv,v典兩vm= u +v其 = inf兵具A共vm−v兲,vm−v典 +␣n具Bv,v典兩v 苸 H其

= inf兵具Avm,vm典 − 2 Re具Avm,v典 + 具Av,v典 +␣n具Bv,v典兩v 苸 H其 艋 具Avm,vm典

− 2 Re具Avm,v_n典 + 具Avn,v_n典 +␣n具Bvn,v_n典 = 1 − 1

m− 2 + 2 m+ 1 −1

n+␣n具Bvn,v_n典

= 1 m−1

n+␣n具Bvn,vn典 → 1 m⬍1

2 as n→ ⬁.

On the other hand,

具Avm,Av_m典 = 1 − 1 m艌1

2. Hence, we have strict inequality in共4.5兲.

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