Journal of Physics Conference Series

Some properties of effect algebra

Wei Li, Wen Han, Yiyue Sun, Jiangtao Li, Jiaying Wang

Published Aug 1, 2024• 0 citations

Abstract

Abstract As a kind of algebraic structure, effect algebra is the foundation of quantum structure and quantum measurement. Sequential effect algebra is a special type of effect algebra that plays an important role in the fundamental research of Quantum Computation. Therefore, we study the related theory of effect algebra in this paper. Finally, four properties and special examples of effect algebra and sequence effect algebra are presented, as well as three properties of effect algebra level sum.

DOI: 10.1088/1742-6596/2813/1/012020

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Journal of Physics Conference Series

Some properties of effect algebra

Wei Li, Wen Han, Yiyue Sun, Jiangtao Li, Jiaying Wang

Published Aug 1, 2024• 0 citations

Publisher

Abstract

DOI: 10.1088/1742-6596/2813/1/012020

Related papers

Sequential Products of Quantum Measurements

Stanley Gudder

Our basic structure is a finite-dimensional complex Hilbert space $H$ . We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ 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.Our basic structure is a finite-dimensional complex Hilbert space $H$. We point out that the set of effects on $H$ form a convex effect algebra. Although the set of operators on $H$ 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.

arXiv:2108.07925Jun 6, 20072 citations

Topologies on Quantum Effects

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Nov 26, 20125 citations

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A quantum effect is an operator $A$ on a complex Hilbert space $H$ that satisfies $0\leq A\leq I$ , ${\cal E} (H)$ is the set of all quantum effects on $H$ . In 2001, Professor Gudder and Nagy studied the sequential product $A\circ B=A^{1/2}BA^{1/2}$ of $A, B\in {\cal E}(H)$ . In 2005, Professor Gudder asked: Is $A\circ B=A^{1/2}BA^{1/2}$ the only sequential product on ${\cal E} (H)$ ? 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 ${\cal E} (H)$ ; secondly, we present a general method for constructing sequential products on ${\cal E} (H)$ ; finally, we study some properties of the sequential products constructed by the methodA quantum effect is an operator $A$ on a complex Hilbert space $H$ that satisfies $0\leq A\leq I$, ${\cal E} (H)$ is the set of all quantum effects on $H$. In 2001, Professor Gudder and Nagy studied the sequential product $A\circ B=A^{1/2}BA^{1/2}$ of $A, B\in {\cal E}(H)$. In 2005, Professor Gudder asked: Is $A\circ B=A^{1/2}BA^{1/2}$ the only sequential product on ${\cal E} (H)$? 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 ${\cal E} (H)$; secondly, we present a general method for constructing sequential products on ${\cal E} (H)$; finally, we study some properties of the sequential products constructed by the method

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Some algebraic properties of the ◇-products of positive operators

BI Hong-mei

Journal of Chongqing Technology and Business UniversityJan 1, 20090 citations

Our basic structure is a finite-dimensional complex Hilbert space

H

. We point out that the set of effects on

H

form a convex effect algebra. Although the set of operators on

H

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

A

on a complex Hilbert space

H

that satisfies

0\leq A\leq I

{\cal E} (H)

is the set of all quantum effects on

H

. In 2001, Professor Gudder and Nagy studied the sequential product

A\circ B=A^{1/2}BA^{1/2}

A, B\in {\cal E}(H)

. In 2005, Professor Gudder asked: Is

A\circ B=A^{1/2}BA^{1/2}

the only sequential product on

{\cal 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

{\cal E} (H)

; secondly, we present a general method for constructing sequential products on

{\cal E} (H)

; finally, we study some properties of the sequential products constructed by the method