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Journal of Physics A Mathematical and Theoretical

Characterizing sequential isomorphisms on Hilbert-space effect algebras

Jinchuan Hou, Kan He, Xiaofei Qi

Published Jul 6, 2010• 9 citations
Publisher

Abstract

Let ⋆ be any sequential product on the Hilbert-space effect algebra with , and be a bijective map. We show that if Φ satisfies Φ( A ⋆ B ) = Φ( A )⋆Φ( B ) for , then there is either a unitary or an anti-unitary operator U such that Φ( A ) = UAU † for every . Let be a Borel function satisfying g (0) = 0, g (1) = 1 and let us define a binary operation ◊ g on by A ◊ g B = A 1/2 g ( A ) Bg ( A )† A 1/2 , where T † denotes the conjugate of the operator T . We also show that a bijective map satisfies Φ( A ◊ g B ) = Φ( A )◊ g Φ( B ) for if and only if there is either a unitary or an anti-unitary operator U such that Φ( A ) = UAU † for every .

DOI
10.1088/1751-8113/43/31/315206

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Let ⋆ be any sequential product on the Hilbert-space effect algebra with , and be a bijective map. We show that if Φ satisfies Φ( A ⋆ B ) = Φ( A )⋆Φ( B ) for , then there is either a unitary or an anti-unitary operator U such that Φ( A ) = UAU † for every . Let be a Borel function satisfying g (0) = 0, g (1) = 1 and let us define a binary operation ◊ g on by A ◊ g B = A 1/2 g ( A ) Bg ( A )† A 1/2 , where T † denotes the conjugate of the operator T . We also show that a bijective map satisfies Φ( A ◊ g B ) = Φ( A )◊ g Φ( B ) for if and only if there is either a unitary or an anti-unitary operator U such that Φ( A ) = UAU † for every .
Let H be a Hilbert space and E ( H ) the effect algebra on H . A bijective map is called an ortho-order automorphism of E ( H ) if for every we have and . The classical theorem of Ludwig states that every such ϕ is of the form , , for some unitary or antiunitary operator U . It is also known that each bijective map on E ( H ) preserving order and coexistency in both directions is of the same form. Can we improve these two theorems by relaxing the bijectivity assumption and/or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion? For both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results.
A quantum effect is an operator A on a complex Hilbert space H that satisfies 0⩽A⩽I. We denote the set of quantum effects by E(H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B∈E(H) by A∘B=A1/2BA1/2. The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A∘B imply that A and B commute for the usual operator product. For example, if A∘B satisfies certain distributive or associative laws, then AB=BA. Moreover, if A∘B∈P(H), then AB=BA and A∘B=B or B∘A=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied.
A quantum effect is an operator AAA on a complex Hilbert space HHH that satisfies 0≤A≤I0\leq A\leq I0≤A≤I, E(H){\cal E} (H)E(H) is the set of all quantum effects on HHH. In 2001, Professor Gudder and Nagy studied the sequential product A∘B=A1/2BA1/2A\circ B=A^{1/2}BA^{1/2}A∘B=A1/2BA1/2 of A,B∈E(H)A, B\in {\cal E}(H)A,B∈E(H). In 2005, Professor Gudder asked: Is A∘B=A1/2BA1/2A\circ B=A^{1/2}BA^{1/2}A∘B=A1/2BA1/2 the only sequential product on E(H){\cal E} (H)E(H)? Recently, Liu and Wu presented an example to show that the answer is negative. In this paper, firstly, we characterize some algebraic properties of the abstract sequential product on E(H){\cal E} (H)E(H); secondly, we present a general method for constructing sequential products on E(H){\cal E} (H)E(H); finally, we study some properties of the sequential products constructed by the method