A simple sufficient condition for the coexistence of quantum effects
aa r X i v : . [ qu a n t - ph ] M a r A SIMPLE SUFFICIENT CONDITION FOR THECOEXISTENCE OF QUANTUM EFFECTS
TEIKO HEINOSAARI
Abstract.
Two quantum effects are considered coexistent if they can be mea-sured together. It is known that commutativity and comparability are sufficientbut not necessary for the coexistence of two effects. We unify those two condi-tions to a simple but more widely applicable sufficient condition. Introduction
Experimental events are in quantum theory described as effect operators (effectsfor short) acting on a complex Hilbert space H ., i.e., selfadjoint operators E : H →H satisfying 0 ≤ h ψ | Eψ i ≤ ψ ∈ H . We denote by E ( H )the set of all effects. This is a convex set and endowed with the complementation E ⊥ = − E .One of the most important relations in E ( H ) is coexistence [1, 2]. By definition,two effects E and F are coexistent if they can be measured together, i.e., arecontained in the range of a single positive operator valued measure (POVM).The properties of the coexistence relation have been investigated from variousaspects [3, 4, 5, 6, 7], and the coexistence relation has been characterized first insome special pairs of qubit effects [8, 9, 10] and later for all pairs of qubit effects[11, 12, 13]. However, in the general case the coexistence relation has remainrather abstract and unmanageable relation. The purpose of this communicationis present a simple sufficient condition for the coexistence of two effects.Our investigation is organized as follows. First, in Section 2 we recall the twowell-known sufficient conditions for coexistence: commutativity and comparabil-ity. In Section 3 we briefly review the peculiar nature of the order structure of E ( H ) and explain why the sufficient condition given by the infimum of two effects,although reasonable, is not very useful. In Section 4 we present a sufficient condi-tion for coexistence using Jordan products, and this generalizes commutativity. InSection 5 we use the notion of generalized infimum to derive a sufficient conditionfor coexistence that covers both the commutativity and comparability criteria.Finally, our investigation is summarized in Section 6.2. Coexistence, Commutativity and Comparability
In this section we recall some basic definitions and results, all that can be foundin [2].By definition, effects E , . . . , E n ∈ E ( H ) are coexistent if there exists a POVM A such that A ( X ) = E , . . . , A ( X n ) = E n for some outcome sets X , . . . , X n .If we are considering a pair of effects, then the definition of coexistence can be rewritten in two simple ways. First, it is easy to see that two effects E and F occur in the range of some POVM if and only if there exist four effects G , G , G and G such that G + G = E , G + G = FG + G + G + G = . (1)Let us notice that other operators in (1) are determined from E, F and G , e.g., G = E − G .There is a natural partial order on E ( H ); E ≤ F if h ψ | Eψ i ≤ h ψ | F ψ i forall ψ ∈ H . Clearly, E ≤ F if and only if there exists an effect E ′ such that E + E ′ = F . We thus obtain another equivalent formulation of coexistence: E and F are coexistent if and only if there exists an effect G such that G ≤ E , G ≤ F , G + ≥ E + F . (2)It is well-known that two effects are coexistent if they commute. Namely, if
E, F are two effects and EF = F E (COMMU)then we can choose G = EF , G = EF ⊥ , G = E ⊥ F , G = E ⊥ F ⊥ (3)and and the equations in (1) are satisfied. The commutativity of E and F guar-antees that all the four operators in (3) are effects since, for instance, EF = √ EF √ E ≥ E and F , their comparability. If E ≤ F or F ≤ E , then wecan choose either G = E or G = F and (2) clearly holds. To fully benefit fromthis fact, we notice that if (1) holds, then G + G = E ⊥ and G + G = F ⊥ .Hence, E and F are coexistent if and only if E, E ⊥ , F, F ⊥ are coexistent. Thisalso means that the following are equivalent:(i) E and F are coexistent(ii) E ⊥ and F are coexistent(iii) E and F ⊥ are coexistent(iv) E ⊥ and F ⊥ are coexistentThus, taking the complement effects into account, we recover the following well-known sufficient condition for coexistence: two effects E and F are coexistentif E ≤ F or F ≤ E or E ≤ F ⊥ or F ⊥ ≤ E (COMP)Two effects that satisfy (COMP) are often called trivially coexistent .Obviously, the coexistence of some pairs of effects, such as and E , followfrom both (COMMU) and (COMP), but it is easy to see that these conditionshave also separate areas of applicability. For instance, let E, F ∈ E ( H ) be twonon-commuting effects and fix a number 0 < t ≤ . Then also the effects tE + (1 − t ) and tF + (1 − t ) (4) are non-commuting, but this latter pair satisfies (COMP) since( tF + (1 − t ) ) ⊥ ≤ tE + (1 − t ) ⇔ (2 t − ≤ t ( E + F ) . (5)To see an example where (COMMU) holds but (COMP) not, fix an effect E such that E (cid:2) and E (cid:3) (e.g. a non-trivial projection), and two numbers0 < s, t <
1. The effects sE + (1 − s ) E ⊥ and tE + (1 − t ) E ⊥ (6)always commute, but (COMP) holds only if s = t or s = 1 − t .3. Infimum
There is a natural way to generalize (COMP) and obtain a new sufficient condi-tion for coexistence using the infimum of two effects. In a closer look this approachturns out to be very restricted, but it well demonstrates the delicate nature of thecoexistence relation and we therefore look briefly look at it.Suppose that the infimum of two effects E and F exists and is denoted by E ∧ F .By definition, E ∧ F is the greatest of all effects C satisfying C ≤ E and C ≤ F .Therefore, whenever there is an effect G satisfying (2), then also E ∧ F satisfies(2). We thus conclude that if E ∧ F exists, then E and F are coexistent if andonly if E ∧ F ≥ E + F − . (7)We should note that even if E ∧ F would exist, (7) is a useful sufficient criterion forcoexistence only if E ∧ F can be expressed in some explicit form. We recall someresults related to the existence and form of the infimum [14, 15]. For E, F ∈ E ( H ),we denote by P E,F the projection onto the closer of ran ( √ E ) ∩ ran ( √ F ). Theinfimum of a projection and an effect always exists and has an explicit expression.Hence, we can calculate E ∧ P E,F and F ∧ P E,F . Then, E ∧ F exists if and onlyif ( E ∧ P E,F ) ∧ ( F ∧ P E,F ) exists, and in this case E ∧ F = ( E ∧ P E,F ) ∧ ( F ∧ P E,F ) . (8)These results do not yet give an explicit form for E ∧ F , but for dim H < ∞ acomplete solution is known [14]. Then E ∧ F exists if and only if E ∧ P E,F and F ∧ P E,F are comparable. If this is the case, E ∧ F is the smaller of E ∧ P E,F and F ∧ P E,F . We conclude that for dim H < ∞ the infimum can be calculatedwhenever it exists, but for dim H = ∞ the explicit form for the infimum seems tobe lacking.Taking into account the complement effects, we can formulate the followingsufficient condition for coexistence. Proposition 1.
Two effects E and F are coexistent if one of the following con-ditions hold: E ∧ F exists and E ∧ F ≥ E + F − E ∧ F ⊥ exists and E ∧ F ⊥ ≥ E + F ⊥ − E ⊥ ∧ F exists and E ⊥ ∧ F ≥ E ⊥ + F − E ⊥ ∧ F ⊥ exists and E ⊥ ∧ F ⊥ ≥ E ⊥ + F ⊥ − (INF) It is easy to see that (INF) is a generalization of (COMP). If, for instance, E ≤ F , then E ∧ F = E and E ∧ F ≥ E + F − is equivalent to ≥ F , and istherefore true. But (INF) is not a generalization of (COMMU) since the infimumof two commuting effects may not exist [16].In the following example we demonstrate the use of (INF). Example 1.
Suppose that
E, F are effects and E is a multiple of a one-dimensionalprojection. We will show that E and F are coexistent if and only if ran ( E ) ⊆ ran ( √ F ) or E + F ≤ .We can write E = e | ψ ih ψ | for some unit vector ψ ∈ H and number 0 < e ≤ ran ( √ E ) = ran ( E ) = C ψ . There are two alternatives: either ψ / ∈ ran ( √ F ) or ψ ∈ ran ( √ F ). In terms of the projection P E,F this means that either P E,F = 0 or P E,F = | ψ ih ψ | , respectively.Let us first assume that P E,F = 0. Then E ∧ P E,F = F ∧ P E,F = 0 and therefore E ∧ F exists and E ∧ F = 0. The coexistence condition E ∧ F ≥ E + F − isequivalent to E + F ≤ .Let us then assume that P E,F = | ψ ih ψ | . We have E ∧ P E,F = h ψ | Eψ i | ψ ih ψ | , F ∧ P E,F = h ψ | F ψ i | ψ ih ψ | . (9)It follows that E ∧ F exists and E ∧ F = min { h ψ | Eψ i , h ψ | F ψ i}| ψ ih ψ | . (10)It is straightforward to verify that the coexistence condition E ∧ F ≥ E + F − is always satisfied.The limited applicability of (INF) originates from the fact that the infimumof two effects exists only rarely. This has become clear from various examplespresented in earlier studies on the infimum. For instance, for dim H = 2 theinfimum of two effects E and F exists if and only if E and F are comparable, orone of them is a multiple of a 1-dimensional projection [17]. As another example,if effects E and F are invertible operators, then E ∧ F exists if and only if E and F are comparable [18]. 4. Jordan product
We will next seek a generalization of commutativity into a more widely appli-cable sufficient condition for coexistence. For
E, F ∈ E ( H ), we denote E ◦ F = ( EF + F E ) . This is called the
Jordan product of E and F . It is clear that E ◦ F is always aselfadjoint operator and E ◦ F ≤ k E k k F k I ≤ I .
However, E ◦ F need not be a positive operator.It is easy to verify that E ◦ F + E ◦ F ⊥ = E , E ◦ F + E ⊥ ◦ F = FE ◦ F + E ⊥ ◦ F + E ◦ F ⊥ + E ⊥ ◦ F ⊥ = I .
Hence, by choosing G = E ◦ F , G = E ◦ F ⊥ , G = E ⊥ ◦ F and G = E ⊥ ◦ F ⊥ , then (1) holds for any choice of E, F ∈ E ( H ). The remaining step forthe coexistence of E and F is to guarantee that these four operators are positive.We thus conclude the following sufficient criterion for coexistence. Proposition 2.
Two effects E and F are coexistent if E ◦ F ≥ E ⊥ ◦ F ≥ E ◦ F ⊥ ≥ E ⊥ ◦ F ⊥ ≥ . (JOR)This condition is a generalization of the commutativity condition (COMMU).Namely, if EF = F E , then E ◦ F = EF = √ EF √ E ≥
0. Therefore, (JOR)holds whenever EF = F E . But (JOR) need not hold if (COMP) holds. Forinstance, let ψ, ϕ ∈ H be two unit vectors such that r := h ψ | ϕ i ∈ R , 0 < r < E = | ψ ih ψ | , F = | ϕ ih ϕ | . Then E + F ≤ , but E ◦ F (cid:3) E ◦ F = r ( | ψ ih ϕ | + | ϕ ih ψ | ) and the eigenvalues of this operator are r (1 + r ) > r ( r − < Example 2.
Let d < ∞ , H = C d and fix an orthonormal basis { ϕ x } d − x =0 for H .We define a unit vector ψ ∈ H as ψ = 1 √ d d − X x =0 ϕ x . Then, we fix a number 0 ≤ λ ≤ E and F by E = λ | ϕ ih ϕ | + (1 − λ ) 1 d , F = λ | ψ ih ψ | + (1 − λ ) 1 d . (11)These two effects commute only if λ = 0. As shown in [19], E and F are guaranteedto be coexistent if λ ≤
12 + √ d − d − ≡ λ MAX ( d ) . (12)We have E ◦ F = λ √ d (cid:0) | ϕ ih ψ | + | ψ ih ϕ | (cid:1) + (13) λ (1 − λ ) d (cid:0) | ϕ ih ϕ | + | ψ ih ψ | (cid:1) + (1 − λ ) d , (14)and the requirement E ◦ F ≥ λ ≤ √ d − d + 4 d − ≡ λ JOR ( d ) . (15)A similar calculation on the three other operator inequalities in (JOR) shows thatthey are less restrictive than E ◦ F ≥
0, hence (JOR) is equivalent to (15). It isstraightforward to confirm that λ JOR (2) = λ MAX (2) and 0 < λ
JOR ( d ) < λ MAX ( d )whenever d ≥
3. We conclude that (JOR) is a more general sufficient conditionfor coexistence than commutativity.
Finally, we note that Proposition 2 has a generalization that gives a sufficientcondition for the coexistence of any finite number of effects. For an effect E , wedenote E (1) = E and E (2) = E ⊥ . Let E , . . . , E n be a finite collection of effects.We define G i i ··· i n = 1 n ! (cid:0) E ( i )1 E ( i )2 · · · E ( i n ) n + E ( i )2 E ( i )1 · · · E ( i n ) n (16)+ all other permutations (cid:1) . It is easy to verify that X i , ··· ,i n ∈{ , } G i ··· i n = E (17)and similarly for other indices. We thus conclude the following generalization ofProposition 2. Proposition 3.
Effects E , . . . , E n are coexistent if G i i ··· i n ≥ for all i , i , · · · , i n ∈{ , } . Example 3.
Let E = ( + e · σ ) , E = ( + e · σ ) , E = ( + e · σ ) . for some orthogonal vectors e , e , e with k e i k ≤
1. The smallest eigenvalueof each operator G , . . . , G is (1 − q k e k + k e k + k e k ). Therefore, werecover the sufficient condition found in [8]: the effects E , E , E are coexistent if k e k + k e k + k e k ≤ . (18)A similar calculation can be done for a non-orthogonal triplet, but then the posi-tivity conditions for G i i i are more complicated.5. Generalized infimum
Since (INF) has a very limited area of applicability, we will seek another wayto generalize (COMP) to a wider sufficient condition for coexistence. For
E, F ∈E ( H ), we denote E ⊓ F = ( E + F − | E − F | ) (19)and call this operator generalized infimum of E and F . The operator E ⊓ F hasmany useful properties similar to the infimum. For instance, if E ≤ F , then E ⊓ F = E . The properties of E ⊓ F have been investigated in various works[18, 20, 21, 22], but here we only need some basic facts.The operator E ⊓ F is selfadjoint and satisfies E ⊓ F ≤ E and E ⊓ F ≤ F . (20)The validity of these operator inequalities can be seen as follows [21]. First, wehave E − F ≤ | E − F | . It follows that E + F − | E − F | ≤ F , which means that E ⊓ F ≤ F . Since | E − F | = | F − E | , a similar argument gives E ⊓ F ≤ E .We look for a sufficient condition for coexistence. If we want that G = E ⊓ F isan effect and satisfies (2), then (20) holds for all pairs but we need to additionallyrequire that E ⊓ F ≥ E ⊓ F ≥ E + F − . (21) The latter operator inequality in (21) can be written in an equivalent form: E ⊓ F ≥ E + F − ⇔ ( E + F − | E − F | ) ≥ E + F − ⇔ (2 − E − F + | E − F | ) ≥ ⇔ E ⊥ ⊓ F ⊥ ≥ . Finally, taking into account the complement effects we conclude the followingsufficient condition for coexistence.
Proposition 4.
Two effects E and F are coexistent if (cid:0) E ⊓ F ≥ E ⊥ ⊓ F ⊥ ≥ (cid:1) or (cid:0) E ⊥ ⊓ F ≥ E ⊓ F ⊥ ≥ (cid:1) . (GINF)It is clear that (COMP) implies (GINF). For instance, if E ≤ F , then E ⊓ F = E ≥ E ⊥ ⊓ F ⊥ = F ⊥ ≥
0. More interestingly, (JOR) implies (GINF), as wenext prove.
Proposition 5.
Let
E, F ∈ E ( H ) . If (JOR) holds, then (GINF) also holds.Proof. We first note that E ◦ F ≥ E − F ) ≤ ( E + F ) . Since thesquare root is an operator monotone function (see e.g. [23]), the latter operatorinequality implies that | E − F | ≤ E + F . This is equivalent to E ⊓ F ≥ (cid:3) Since (COMMU) implies (JOR), we conclude that (GINF) covers both (COMP)and (COMMU). But (GINF) has a wider area of applicability than (COMP) and(COMMU) together. This is demonstrated in the following example.
Example 4.
Let H = C and denote σ = ( σ x , σ y , σ z ), where σ x , σ y and σ z arethe usual Pauli matrices. We consider two effects E and F of the form E = ( + e · σ ) , F = ( + f · σ ) (22)for some e , f ∈ R with k e k ≤ , k f k ≤
1. It was first proved in [8] that E and F are coexistent if and only if k e + f k + k e − f k ≤ . (23)A direct calculation shows that the smallest eigenvalue of E ⊓ F , E ⊓ F ⊥ , E ⊥ ⊓ F and E ⊥ ⊓ F ⊥ is (2 − k e − f k − k e + f k ). We conclude that (GINF) is equivalentto (23), hence necessary and sufficient for the coexistence of E and F .The following example shows that (GINF) is not a necessary condition for co-existence. Example 5.
Let E = ( + e · σ ) , F = ( β + f · σ ) (24)for some for some orthogonal vectors e , f ∈ R with k e k ≤ , k f k ≤
1, and k f k ≤ β ≤ − k f k . It was first proved in [10] that E and F are coexistent if andonly if 2 k e k ≤ q β − k f k + q (2 − β ) − k f k . (25) The choices k e k = k f k = 2 / β = 3 / E ⊓ F and E ⊥ ⊓ F have a negative eigenvalue.We conclude that (GINF) is not a necessary condition for coexistence.Finally, let us consider the relationship between (GINF) and (INF). It is easyto see that (GINF) does not imply (INF). Namely, (GINF) can hold even if noneof E ∧ F , E ∧ F ⊥ , E ⊥ ∧ F , E ⊥ ∧ F ⊥ exist. This can be seen from Example 4and the result mentioned earlier [17]: the infimum of E, F ∈ E ( C ) exists if andonly if E and F are comparable or one of them is a multiple of a one-dimensionalprojection.An interesting question is whether (INF) implies (GINF), and if not, to finda sufficient condition that covers both of them. We cannot offer a solution atthe moment, but there is some indication that (INF) may imply (GINF). It wasproved in [21] that if E ⊓ F ≥ E ∧ F exists, then E ∧ F = E ⊓ F . Inthis case, the operator inequalities E ∧ F ≥ E + F − and E ⊥ ⊓ F ⊥ ≥ E ⊓ F ≥
0, and this would thenmean that (INF) implies (GINF). Related to this conjecture, we recall that for aneffect E and a projection P , EP = P E if and only if E ⊓ P ≥ E ∧ P always exists [14], we see that (INF) holds if and only if EP = P E . Therefore, (INF) and (GINF) are equivalent conditions for two effectsif one of them is a projection. 6.
Summary
There are two well-known sufficient criteria for coexistence, comparability andcommutativity. We have introduced a new sufficient condition that is a general-ization of both comparability and commutativity. For two effects E and F , thegeneralized infimum is defined as E ⊓ F = ( E + F − | E − F | ). The new sufficientcondition is (cid:0) E ⊓ F ≥ E ⊥ ⊓ F ⊥ ≥ (cid:1) or (cid:0) E ⊥ ⊓ F ≥ E ⊓ F ⊥ ≥ (cid:1) . We have demonstrated that this new condition does not only combine compara-bility and commutativity, but covers also other pairs of coexistent effects. It is,however, not a necessary condition for coexistence.There are some natural questions related to the above sufficient condition. Isit possible to write it in an equivalent but simpler way, for instance by droppingsome of the operator inequalities? Is the infimum condition (INF) introduced inSec. 3 covered by the new condition?
Acknowledgements
This work has been supported by the Academy of Finland (grant no. 138135).
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