On the infimum of quantum effects
Aurelian Gheondeaa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 Gudderb兲
Department of Mathematics, University of Denver, Denver, Colorado 80208 Peter Jonasc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
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A1/2兲艚Ran共B1/2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∧PA,Band B∧PA,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∧PA,B and B∧PA,B are not comparable. In addition, we prove that,
assuming A∧B exists, PA,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 PA,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 effectA with its values 1 and 0 reversed. We also present a simple spectral characterization of when A∧A
⬘
exists inE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∧PA,B and B∧PA,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∧PA,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∧PA,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 algebra7,10 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
T X Y Z册
=冋
I 0 YT−1 I册冋
T 0 0 Z − YT−1X册冋
I T−1X 0 I册
. 共2.1兲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
A1 A1 1/2⌫A 2 1/2 A21/2⌫*A11/2 A2册
with respect toH = H1丣H2, 共2.2兲 where A1苸B共H1兲+, A2苸B共H2兲+, and⌫苸B共H2,H1兲 is contractive.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A
1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A
1兲, then A∧P =
冋
A1 1/2共I − ⌫⌫*兲A 1 1/2 00 0
册
with respect toH = Ran共P兲丣Ker共P兲. 共2.3兲Proof: Let A苸E共H兲 and P苸P共H兲. In the following we consider the orthogonal
decomposi-tion H=Ran共P兲丣Ker共P兲. By Theorem 2.1 A has a matrix representation as in 共2.2兲, with A1 苸B共Ran共P兲兲+, A
2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
A1 1/2⌫⌫*A 1 1/2 A 1 1/2⌫A 2 1/2 A21/2⌫*A11/2 A2册
=关⌫*A 1 1/2A 2 1/2兴*关⌫*A 1 1/2A 2 1/2兴 艌 0, 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
C1 00 0
册
with respect toH = Ran共P兲丣Ker共P兲. Then0艋 A − C =
冋
A1− C1 A1 1/2⌫A 2 1/2 A21/2⌫*A11/2 A2册
. 共2.4兲The matrix with operator entries in共2.4兲 can be factored as
冋
IRan共P兲 0 0 A21/2册
冋
A1− C1 A1 1/2⌫ ⌫*A 1 1/2 I Ker共P兲册
冋
IRan共P兲 0 0 A21/2册
. 共2.5兲Note that by Ker共⌫兲傶Ker共A2兲 or, equivalently, Ran共⌫*兲債Ran共A
2兲, A−C and each of the factors
of共2.5兲 map the subspace H
⬘
= Ran共P兲丣Ran共A2兲 into itself. Since diag共IRan共P兲A21/2兲 regarded as anoperator 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 A11/2⌫⌫*A11/2艋A1− C1, that is, C1艋A11/2共I
Ran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∧EA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
A BB* I
册
共2.6兲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A1/2兲艚Ran共B1/2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∧PA,B and
B∧PA,B are comparable, and that A∧B is the smaller of A∧PA,B and B∧PA,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∧PA,Band B∧PA,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∧PA,B兲∧共B∧PA,B兲 exists.
In this case A∧B =共A∧PA,B兲∧共B∧PA,B兲.
Proof: Note first that the operators A∧PA,Band B∧PA,Bexist, e.g., by Theorem 2.2.
Let us assume that 共A∧PA,B兲∧共B∧PA,B兲 exists and let C苸E共H兲 be such that C艋A,B, thus
we have Ran共C1/2兲債Ran共A1/2兲艚Ran共B1/2兲債Ran共PA,B兲 and hence C艋 PA,B. Therefore, C
艋A∧PA,B and C艋B∧PA,B and hence, by the majorization theorem as in Ref. 6, C
艋共A∧PA,B兲∧共B∧PA,B兲. Taking into account that 共A∧PA,B兲∧共B∧PA,B兲艋A,B it follows that A∧B
exists and equals共A∧PA,B兲∧共B∧PA,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∧PA,B, A∧B艋B∧PA,B. Let C苸E共H兲 be such that C艋A∧PA,B, B∧PA,B. Then C
艋A, B, PA,Band, in particular, C艋A∧B. 䊐
One may ask for which orthogonal projections P except PA,Bthe statement of Proposition 2.4
is true. It turns out that PA,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A1/2兲艚Ran共B1/2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 An&A and Bn&B strongly, then An: Bn&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A1/2兲. Similarly we have
Ran共共A∧B兲1/2兲債Ran共B1/2兲, hence Ran共共A∧B兲1/2兲債Ran共A1/2兲艚Ran共B1/2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∧PA,B兲∧共B∧PA,B兲. Evidently, 共A∧PA,B兲∧共B∧PA,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∧PA,B兲∧共B∧PA,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 effectA 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
12, 1
兴
or in关
0 , 1 2兴
艛兵1其,共iii兲 A∧PA,I−A and 共I−A兲∧PA,I−Aare comparable, that is, either A∧PA,I−A艋共I−A兲∧PA,I−Aor 共I−A兲∧PA,I−A艋A∧PA,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
t, 0艋 t 艋 1 2,1 − t, 12艋 t 艋 1,
冎
共3.1兲we have, by continuous functional calculus, A∧共I−A兲=g共A兲.
Proof: Let EA denote the spectral function of A. In view of Proposition 2.4, A∧共I−A兲 exists
if and only if 共A∧PA,I−A兲∧共共I−A兲∧PA,I−A兲 exists. A moment of thought shows that PA,I−A
= EA共共0,1兲兲 and hence, by Remark 2.3, we have that A∧PA,I−A= AEA共共0,1兲兲 and 共I−A兲∧PA,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 EA of A, let E1= EA共关0,1/2兴, A1= A兩E1H, E2= EA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
D1 D1 1/2⌫D 2 1/2 D21/2⌫*D11/2 D2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
D1− A1 D1 1/2⌫D 2 1/2 D21/2⌫*D11/2 D2−共I2− A2兲册
. 共3.2兲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= EA共共,− + 1/2兲兲, E,2= EA共共 + 1/2,1 − 兲兲. 共3.3兲
Denote E= E,1+ E,2and A= A兩EH. 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EH=g共A兲 coincides with A∧共I−A兲. Indeed, assume that for some C苸E共EH兲 we have C艋A, I − A. Then, letting
C = CE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, Dcoincides 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
3兲−1, consider an arbitrary contraction T苸B共E,2H,E,1H兲. In the following all operator matrices are understood with respect to the decomposition E,1H丣E,2H. Then, letting
C =
冋
A冑
,1− dI,1冑
3 dT 3 dT* I ,2− A,2− dI,2册
=冋
A冑
,1−I,1+冑
3 dI,1冑
3dT 3 dT* I ,2− A,2−I,2+冑
3 dI,2册
=冋
A,1−I,1 0 0 I,2− A,2−I,2册
+冑
3d冋
I,1 T T* I,2册
艌 0, we have A− C =冋
dI,1 −冑
3 dT −冑
3 dT* 2A,2− I,2+ dI,2册
=冋
0 0 0 2A,2− I,2− 2 dI,2册
+ d冋
I,1 −冑
3T −冑
3T* 3I,2册
艌 0 and I − A− C =冋
I,1− 2A,1+ dI,1 −冑
3T −冑
3 dT* dI,2册
=冋
I,1− 2A,1− 2 dI,1 0 0 0册
+ d冋
3I,1 −冑
3T −冑
3 dT* I,2册
艌 0.But, the operator
共A∧共I− A兲兲 − C = g共A兲 − C = d
冋
I,1 −
冑
3T −冑
3T* I,2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 fA+Bbe the affine 共that is, linear on convex combinations兲 mapping defined as in Ref. 9 by
fA+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, fA+Bis well defined. Since fA+B is
an affine isomorphism, A∧B exists if and only if fA+B共A兲∧fA+B共B兲 exists. As fA+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 fA共C兲∧fA共D兲 exists and, in this case, we have
fA共C∧D兲 = fA共C兲∧fA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∧PA,Band
B∧PA,B exists or, equivalently, the infimum of the restrictions A˜ ªA∧PA,B兩 PA,BH and B˜
ªB∧PA,B兩 PA,BH exists. Since Ker共A˜+B˜兲=兵0其, A˜∧˜ exists if and only if fB A˜ +B˜共A˜兲∧fA˜ +B˜共B˜兲 exists,
and for the pair fA˜ +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 fA˜ +B˜共A˜兲 and fA˜ +B˜共B˜兲, A˜∧B˜ exists if and
A∧PA,B and B∧PA,B. For a finite dimensional Hilbert space it was proven in Ref. 19 that the
infimum of the operators A∧PA,Band B∧PA,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∧PB1,B2兲 and 共B2∧PB1,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 A1/2, that is,共A1/2x兲共t兲
=兩t兩x共t兲, x苸L2关−1,1兴. Note that A, and hence A1/2, are injective.
Let 1 be the constant function equal to 1 on 关−1,1兴, 共t兲ªsgn共t兲, and ±ª 1
2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
0 0 0 0 1 0 0 0 12I0冥
, C1=冤
1 0 0 0 0 0 0 0 12I0冥
,where I0is the identity operator onH0. Clearly we have C1+ C2= I and letting B1= A1/2C1A1/2, B2= A1/2C2A1/2,
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 PB1,B2= I, that is,
Ran共B11/2兲艚Ran共B21/2兲 is dense in H. We divide the proof in several steps.
Step 1: A1/2H0is 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A1/2h0, f典 = 具h0,A1/2f典.
Then A1/2f is a linear combination of the functions 1 and, that is, there exist scalars␣andsuch
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
␣+ t , 0⬍ t 艋 1, −␣ t , − 1艋 t ⬍ 0.冧
Since f苸L2关−1,1兴 this shows that f =0 and the claim is proven.
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− f1 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− f2 储艋 8. 共4.3兲 Consequently, 储h0− f2 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− f3 储艋储h0− f1 储+储f1− f2 储+储f2− f3 储艋 ,
and the claim is proven.
Finally, we prove the following.
Step 3: PB1,B2= I, that is, Ran共B1
1/2兲艚Ran共B 2
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,
A1/2共A−1/2F0兲 is a linear submanifold in H0and dense in it. Since the restrictions of C1and C2to
H0coincide with 1
2I0, it follows that the linear manifolds C1A1/2共A−1F0兲 and C2A1/2共A−1F0兲
coin-cide and are dense in H0. Consequently, the linear manifolds A1/2C1A1/2共A−1F0兲 and A1/2C
2A1/2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B 2 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∧PA,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:␣n
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A1/2兲艚Ran共B1/2兲 it follows A:nB艋 P
A,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, A1/2is the operator of multiplication with the
independent variable t. Note that both A and A1/2are 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 A1/2. Let C be a non-negative contraction in H with kernel C1 and define B = A1/2CA1/2. Then the operator B is injective and hence its range is dense in H. Since
Ran共B兲債Ran共B1/2兲 and, by construction, Ran共B兲債Ran共A1/2兲 as well, it follows that
Ran共A1/2兲艚Ran共B1/2兲 is dense in H, hence P
A,B= I.
For each n艌1 consider the function vn苸L2关0,1兴 defined by
vn共t兲 =
再
0, 0艋 t 艋 1/n, 1/t 1/n⬍ t 艋 1.
冎
Note that A1/2vn=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,vn典 = 具CA1/2vn,A1/2vn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,vn典 + 具Avn,vn典 +␣n具Bvn,vn典 = 1 − 1 m− 2 + 2 m+ 1 − 1 n+␣n具Bvn,vn典 = 1 m− 1 n+␣n具Bvn,vn典 → 1 m⬍ 1 2 as n→ ⬁. On the other hand,
具Avm,Avm典 = 1 −
1
m艌
1 2. Hence, we have strict inequality in共4.5兲.
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