A Note on Sequential Product of Quantum Effects

Chunyuan Deng

doi:10.1155/2013/520436

Abstract

The quantum effects for a physical system can be described by the set of positive operators on a complex Hilbert space that are bounded above by the identity operator . For , let be the sequential product and let be the Jordan product of . The main purpose of this note is to study some of the algebraic properties of effects. Many of our results show that algebraic conditions on and imply that and have diagonal operator matrix forms with as an orthogonal projection on closed subspace being the common part of and . Moreover, some generalizations of results known in the literature and a number of new results for bounded operators are derived.

Highlights

Let H, B(H), and P(H) be complex Hilbert space, the set of all bounded linear operators on H, and the set of all orthogonal projections on H, respectively
Since R(B) ⊆ R(A), A and B can be written as operator matrices A = A1 ⊕ 0, B = B1 ⊕ 0 with respect to the space decomposition H = R(P) ⊕ N(P), respectively, where A1 is an injective positive operator
In [11, Lemma 3.4], the authors had gotten that if A, B ∈ E(H) and dim H < ∞, A ∘ B + A󸀠 ∘ B = B󸀠 if and only if B = (1/2)I. The authors said they did not know if the condition dim H < ∞ can be relaxed

Summary

Chunyuan Deng

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. The main purpose of this note is to study some of the algebraic properties of effects. Many of our results show that algebraic conditions on A ∘ B and A ∗ B imply that A and B have 3 × 3 diagonal operator matrix forms with IR(A)∩R(B) as an orthogonal projection on closed subspace R(A) ∩ R(B) being the common part of A and B. Some generalizations of results known in the literature and a number of new results for bounded operators are derived

Introduction

Main Results

We have

We get