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arXiv:1509.09110 · Journal of Mathematical Physics

Continuity of the sequential product of sequential quantum effect algebras

Qiang Lei, Xiaochao Su, Junde Wu

Published Apr 1, 2016• 1 citation
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Abstract

In order to study quantum measurement theory, sequential product defined by A∘B = A1/2BA1/2 for any two quantum effects A, B has been introduced. Physically motivated conditions ask the sequential product to be continuous with respect to the strong operator topology. In this paper, we study the continuity problems of the sequential product A∘B = A1/2BA1/2 with respect to other important topologies, such as norm topology, weak operator topology, order topology, and interval topology.

DOI
10.1063/1.4945085

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In order to study quantum measurement theory, sequential product defined by A∘B = A1/2BA1/2 for any two quantum effects A, B has been introduced. Physically motivated conditions ask the sequential product to be continuous with respect to the strong operator topology. In this paper, we study the continuity problems of the sequential product A∘B = A1/2BA1/2 with respect to other important topologies, such as norm topology, weak operator topology, order topology, and interval topology.
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.
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.
Unsharp quantum measurements can be modelled by means of the class E ( H ) \mathcal {E}(\mathcal {H}) of positive contractions on a Hilbert space H \mathcal {H} , in brief, quantum effects. For A , B ∈ E ( H ) A,B\in \mathcal {E}(\mathcal {H}) the operation of sequential product A ∘ B = A 1 / 2 B A 1 / 2 A\circ B=A^{1/2}BA^{1/2} was proposed as a model for sequential quantum measurements. We continue these investigations on sequential product and answer positively the following question: the assumption A ∘ B ≥ B A\circ B\geq B implies A B = B A = B AB=BA=B . Then we propose a geometric approach of quantum effects and their sequential product by means of contractively contained Hilbert spaces and operator ranges. This framework leads us naturally to consider lattice properties of quantum effects, sums and intersections, and to prove that the sequential product is left distributive with respect to the intersection.
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