## Abstract and Applied Analysis

### A Note on Sequential Product of Quantum Effects

Chunyuan Deng

#### Abstract

The quantum effects for a physical system can be described by the set $ℰ\left(ℋ\right)$ of positive operators on a complex Hilbert space $ℋ$ that are bounded above by the identity operator $I$. For $A,B\in ℰ\left(ℋ\right)$, let $A\circ B={A}^{1/2}B{A}^{1/2}$ be the sequential product and let $A\mathrm{*}B=\left(AB+BA\right)/\mathrm{2}$ be the Jordan product of $A,B\in ℰ\left(ℋ\right)$. 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 $A\circ B$ and $A\mathrm{*}B$ imply that $A$ and $B$ have $\mathrm{3}×\mathrm{3}$ diagonal operator matrix forms with ${I}_{\overline{ℛ\left(A\right)}\cap \overline{ℛ\left(B\right)}}$ as an orthogonal projection on closed subspace $\overline{ℛ\left(A\right)}\cap \overline{ℛ\left(B\right)}$ being the common part of $A$ and $B$. Moreover, some generalizations of results known in the literature and a number of new results for bounded operators are derived.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 520436, 6 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512024

Digital Object Identifier
doi:10.1155/2013/520436

Mathematical Reviews number (MathSciNet)
MR3096824

Zentralblatt MATH identifier
1364.81152

#### Citation

Deng, Chunyuan. A Note on Sequential Product of Quantum Effects. Abstr. Appl. Anal. 2013 (2013), Article ID 520436, 6 pages. doi:10.1155/2013/520436. https://projecteuclid.org/euclid.aaa/1393512024

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