aa r X i v : . [ m a t h - ph ] S e p Remarks on the Sequential Products ∗ Liu Weihua, Wu Zhaoqi, Wu Junde † Department of Mathematics, Zhejiang University, Hangzhou 310027, P. R. China
Abstract . In this paper, we show that those sequential products which were proposed by Liu and Shenand Wu in [J. Phys. A: Math. Theor. , 185206 (2009), J. Phys. A: Math. Theor. , 345203 (2009)]are just unitary equivalent to the sequential product A ◦ B = A BA . Key words.
Hilbert space, L¨uders operation, Sequential product . Pacs. L be a quantum-mechanical system and it be represented by a complex Hilbert space H .Each self-adjoint operator A on H satisfies that 0 ≤ A ≤ I is said to be a quantum effect ([1-2]).Quantum effects represent yes-no measurements that may be unsharp. The set of quantum effectson H is denoted by E ( H ). The subset P ( H ) of E ( H ) consisting of orthogonal projection operatorsrepresents sharp yes-no measurements. Let T ( H ) be the set of trace class operators on H and S ( H ) the set of density operators on H , i.e., the state set of quantum system L .As we knew, a quantum measurement can be described as a quantum operation whichis a completely positive linear mapping Φ : T ( H ) → T ( H ) such that for each T ∈ S ( H ), ∗ This project is supported by Zhejiang Innovation Program for Graduates (YK2009002) and Natural ScienceFoundation of China (10771191 and 10471124) and Natural Science Foundation of Zhejiang Province of China(Y6090105). † Corresponding author E-mail: [email protected] ≤ tr [Φ( T )] ≤ P ∈ P ( H ), the so-called L¨uders operation Φ PL is defined by T → P T P , in physics, it implied that if the quantum-mechanical system L is in state W ∈ S ( H ), thenthe probability that the measurement P is observed is given by p W ( P ) = tr ( P W P ), moreover,the resulting state after the measurement P is observed is P W Ptr ( P W P ) whenever tr ( P W P ) = 0([4]). Each quantum effect B ∈ E ( H ) gives to a general L¨uders operation Φ BL : T → B T B . If A, B ∈ E ( H ), then the composition operation Φ BL ◦ Φ AL defines a new operation and is called asequential operation as it is obtained by performing first Φ AL and then Φ BL . It is easily to provethat Φ BL ◦ Φ AL = Φ A BA L ([5 , P − ]). Let us denote A BA by A ◦ B , then A ◦ B ∈ E ( H ) and ◦ has the following important properties ([6-7]):(S1). The map B → A ◦ B is additive for each A ∈ E ( H ), that is, if B + C ≤ I ,then ( A ◦ B ) + ( A ◦ C ) ≤ I and ( A ◦ B ) + ( A ◦ C ) = A ◦ ( B + C ).(S2). I ◦ A = A for all A ∈ E ( H ).(S3). If A ◦ B = 0, then A ◦ B = B ◦ A .(S4). If A ◦ B = B ◦ A , then A ◦ ( I − B ) = ( I − B ) ◦ A and A ◦ ( B ◦ C ) = ( A ◦ B ) ◦ C for all C ∈ E ( H ).(S5). If C ◦ A = A ◦ C , C ◦ B = B ◦ C , then C ◦ ( A ◦ B ) = ( A ◦ B ) ◦ C and C ◦ ( A + B ) = ( A + B ) ◦ C whenever A + B ≤ I .Professor Gudder called A ◦ B the sequential product of A and B , it represents the quantumeffect produced by first measuring A then measuring B ([6-7]). In [8], Gudder asked: is A ◦ B = A BA the only operation on E ( H ) which satisfies the properties (S1)-(S5) ? In [9], Liu andWu showed that if H is a finite dimensional complex Hilbert space, f z ( u ) is the complex-valuedfunction defined on [0 , f z ( u ) = exp z (ln u ) if u ∈ (0 ,
1] and f z (0) = 0, and denote A i = f i ( A ) , A − i = f − i ( A ), then A ◦ B = A / A i BA − i A / defined a new sequential productwhich satisfies the properties (S1)-(S5), thus, Gudder’s problem was answered negatively.Note that the sequential product A ◦ B = A BA = A B ( A ) ∗ of A and B can onlydescribe the instantaneous measurement, that is, the measurement B is completed at once afterthe measurement A is performed. In order to describe a more complicated process where we allowa duration between the measurement A with the measurement B , then we need to replace A with f ( A ), ( A ) ∗ with ( f ( A )) ∗ , where f ( A ) is a function of A which describe the change of A was made by the duration between B with A . Thus, we need to consider the following generalsequential product f ( A ) B ( f ( A )) ∗ .By the above motivation, in [10], Shen and Wu proved the following result: Theorem 1 . Let H be a finite dimensional complex Hilbert space, for each A ∈ E ( H ), sp ( A ) the spectra of A and B ( sp ( A )) the set of all bounded complex Borel functions on sp ( A ).Take a f A ∈ B ( sp ( A )). Define A ⋄ B = f A ( A ) B ( f A ( A )) ∗ for B ∈ E ( H ). Then ⋄ has the properties(S1)-(S5) iff the set { f A } A ∈E ( H ) satisfies the following conditions:(i) For every A ∈ E ( H ) and t ∈ sp ( A ), | f A ( t ) | = √ t ;2ii) For any A, B ∈ E ( H ), if AB = BA , then there exists a complex constant ξ such that | ξ | = 1 and f A ( A ) f B ( B ) = ξf AB ( AB ).Note that for each A ∈ E ( H ), we can take many f A ∈ B ( sp ( A )) satisfies the conditions (i)and (ii), so, Theorem 1 told us that for each given finite dimensional complex Hilbert space H ,there are many sequential products on ( E ( H ) , , I, ⊕ ).In this note, we show that these sequential products are unitary equivalent to the sequentialproduct A ◦ B = A BA .Firstly, we need the following: Lemma 1.1 ([10]) . If { f A } A ∈E ( H ) satisfies the conditions (i) and (ii) of Theorem 1, thenwe have(1) f A ( A ) f A ( A ) = f A ( A ) f A ( A ) = A , ( f A ( A )) ∗ = f A ( A ).(2) If 0 ∈ sp ( A ), then f A (0) = 0.(3) If A = n P k =1 λ k E k , where { E k } nk =1 are pairwise orthogonal projections and λ k = 0, then f A ( A ) = n P k =1 f A ( λ k ) E k .Our main result is: Theorem 2 . Let H be a finite dimensional complex Hilbert space. Then the sequentialproduct f A ( A ) B ( f A ( A )) ∗ on ( E ( H ) , , I, ⊕ ) is unitary equivalent to the sequential product A ◦ B = A BA . Proof . Let A = n P k =1 λ k E k be the spectra decomposition of A , where { E k } nk =1 be pairwiseorthogonal projection operators and λ k > , k = 1 , , · · · , n . By condition (i) of Theorem 1, wehave | f A ( λ k ) | = √ λ k , so f A ( λ k ) = √ λ k e iθ k for some real number θ . Let E = I − P nk =1 E k and U = P nk =1 e iθ k E k + E . Then U is an unitary operator and it is easy to see that AU = U A , soby Lemma 1.1, we have f A ( A ) = A / U . Thus, A ⋄ B = f A ( A ) Bf A ( A ) = A / U B ( A / U ) ∗ = U ( A / BA / ) U ∗ = U ( A ◦ B ) U ∗ and the conclusion is proved. References [1] G. Ludwig,
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