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Journal of Northwest University

Horizontal sums of effect algebras and the uniqueness of sequential products

Cao Huai-xin

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

Aim To introduce the logical definition of the horizontal sum of effect algebras and study the uniqueness of sequential products on effect algebra.Methods Using the definition of effect algebra,sequential effect algebra and the polar decomposition of operator.Results There exists an infinite number of sequential products on the horizontal sum HS(e(H),) of the Hilbert space effect algebra e(H) and .Conclusion An effect algebra admits an infinite number of sequential products.As a special effect algebra,orthoalgebra can be decomposed to the horizontal sum of sub-effect algebra.

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Aim To introduce the logical definition of the horizontal sum of effect algebras and study the uniqueness of sequential products on effect algebra.Methods Using the definition of effect algebra,sequential effect algebra and the polar decomposition of operator.Results There exists an infinite number of sequential products on the horizontal sum HS(e(H),) of the Hilbert space effect algebra e(H) and .Conclusion An effect algebra admits an infinite number of sequential products.As a special effect algebra,orthoalgebra can be decomposed to the horizontal sum of sub-effect algebra.
Our basic structure is a finite-dimensional complex Hilbert space HHH. We point out that the set of effects on HHH form a convex effect algebra. Although the set of operators on HHH also form a convex effect algebra, they have a more detailed structure. W introduce sequential products of effect and operations. Although these have already been studied, we introduce the new concept of sequential products of effects with operations and operations with effects. We then consider various special types of operations. After developing properties of these concepts, the results are generalized to include observables and instruments. In particular, sequential products of observables with instruments and instruments with observables are developed. Finally, we consider conditioning and coexistence of observables and instruments.
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
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 sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C ∗ -algebras. A SEA is called normal when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition convex , i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when 0=1 .We show that any normal SEA E splits as a direct sum E=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb , a convex normal SEA Ec , and a newly identified type of normal SEA Eac we dub purely almost-convex .Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.
The aim of this paper is to show that there can be either only one or uncountably many contexts in any spectral effect algebra, answering a question posed in [S. Gudder, Convex and Sequential Effect Algebras, (2018), arXiv:1802.01265]. We also provide some results on the structure of spectral effect algebras and their state spaces and investigate the direct products and direct convex sums of spectral effect algebras. In the case of spectral effect algebras with sharply determining state space, stronger properties can be proved: the spectral decompositions are essentially unique, the algebra is sharply dominating and the set of its sharp elements is an orthomodular lattice. The article also contains a list of open questions that might provide interesting future research directions.