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Journal of Ankang University

The Sequential Product of Quantum Effect

Chunfang Shao

Published Jan 1, 2009• 2 citations
OpenAlex

Abstract

For every A,B or A,B P(H),we defined the sequential product of A,B by AB= ABA.The main purpose of this article is to investigate some of the algebraic properties about the sequential product of quantum effect.Put some algebraic conditions on AB imply that A and B commute for the usual operator product.The result which S.Gudder got in(4),we give a wholely new proof by the technique of block operator matrix which makes the geometry structure of operators more clear.

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For every A,B or A,B P(H),we defined the sequential product of A,B by AB= ABA.The main purpose of this article is to investigate some of the algebraic properties about the sequential product of quantum effect.Put some algebraic conditions on AB imply that A and B commute for the usual operator product.The result which S.Gudder got in(4),we give a wholely new proof by the technique of block operator matrix which makes the geometry structure of operators more clear.
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.
It is assumed that e(H) is the set of all positive contraction operators,and P(H) is the set of all orthogonal projections in Hilbert space H in this paper.Forthermore,A∈e(H) is called a quantum effect,called for short.If A,B∈e(H),A°B=A12BA12,is called the sequential product of effect A and B.By using the technique of operator block,the equivalent conditions of A°B∈P(H)when A,B∈e(H) or A,B∈P(H) are discussed respectively.Based on the theory of space decomposition,its geometry structure is further made more clear to be convenient to do research on its corresponding properties.
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
The quantum effects for a physical system can be described by the set E(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. For A, B∊E(H), the operation of sequential product A∘B=A1∕2BA1∕2 was proposed as a model for sequential quantum measurements. A nice investigation of properties of the sequential product has been carried over [Gudder, S. and Nagy, G., “Sequential quantum measurements,” J. Math. Phys. 42, 5212 (2001)]. In this note, we extend some results of this reference. In particular, a gap in the proof of Theorem 3.2 in this reference is overcome. In addition, some properties of generalized infimum A⨅B are studied.
It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product a◦b=aba on an operator algebra. We give three additional properties, each of which characterises the standard sequential product on either a von Neumann algebra or a Euclidean Jordan algebra. These properties are (1) invariance under the application of unital order isomorphisms, (2) symmetry of the sequential product with respect to a certain inner product, and (3) preservation of invertibility of the effects. To give these characterisations, we first have to study convex σ-sequential effect algebras. We show that these objects correspond to unit intervals of spectral order unit spaces with a homogeneous positive cone.