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arXiv:0903.5115

The average value inequality in sequential effect algebras

Jun Shen, Junde Wu

Published Mar 30, 2009• 0 citations
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Abstract

A sequential effect algebra $(E,0,1, \oplus, \circ)$ is an effect algebra on which a sequential product $\circ$ with certain physics properties is defined, in particular, sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Gudder asked the following problem: If $a, b\in (E,0,1,\oplus, \circ)$ and $a\bot b$ and $a\circ b\bot a\circ b$, is it the case that $2(a\circ b)\leq a^2\oplus b^2$ ? In this paper, we construct an example to answer the problem negatively.

DOI
10.48550/arxiv.0903.5115

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A sequential effect algebra (E,0,1,⊕,∘)(E,0,1, \oplus, \circ)(E,0,1,⊕,∘) is an effect algebra on which a sequential product ∘\circ∘ with certain physics properties is defined, in particular, sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Gudder asked the following problem: If a,b∈(E,0,1,⊕,∘)a, b\in (E,0,1,\oplus, \circ)a,b∈(E,0,1,⊕,∘) and a⊥ba\bot ba⊥b and a∘b⊥a∘ba\circ b\bot a\circ ba∘b⊥a∘b, is it the case that 2(a∘b)≤a2⊕b22(a\circ b)\leq a^2\oplus b^22(a∘b)≤a2⊕b2 ? In this paper, we construct an example to answer the problem negatively.
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
Sequential effect algebra is an important model for studying quantum measurement theory. In 2005, Professor Gudder presented 25 open problems to motivate its study. The 20th problem asked: In a sequential effect algebra, if the square root of some element exists, is it unique ? We can strengthen the problem as following: For each given positive integer n>1n>1n>1, is there a sequential effect algebra such that the n-th root of its some element ccc is not unique and the n-th root of ccc is not the k-th root of ccc ($k
Let EEE be an effect algebra and ESE_SES​ be the set of all sharp elements of EEE. EEE is said to be sharply dominating if for each a∈Ea\in Ea∈E there exists a smallest element a^∈Es\widehat{a}\in E_sa∈Es​ such that a≤a^a\leq \widehat{a}a≤a. In 2002, Professors Gudder and Greechie proved that each σ\sigmaσ-sequential effect algebra is sharply dominating. In 2005, Professor Gudder presented 25 open problems in International Journal of Theoretical Physics, Vol. 44, 2199-2205, the 3th problem asked: Is each sequential effect algebra sharply dominating? Now, we construct an example to answer the problem negatively.
Let EEE be an effect algebra and ESE_SES​ be the set of all sharp elements of EEE. EEE is said to be sharply dominating if for each a∈Ea\in Ea∈E there exists a smallest element a^∈Es\widehat{a}\in E_sa∈Es​ such that a≤a^a\leq \widehat{a}a≤a. In 2002, Professors Gudder and Greechie proved that each σσσ-sequential effect algebra is sharply dominating. In 2005, Professor Gudder presented 25 open problems in International Journal of Theoretical Physics, Vol. 44, 2199-2205, the 3th problem asked: Is each sequential effect algebra sharply dominating? Now, we construct an example to answer the problem negatively.
顺序的效果代数学(E， 0， 1，鈯？掳) 是有某些物理性质的顺序的产品掳在上被定义的效果代数学；特别地，顺序的效果代数学是为学习量测量理论的一个重要模型。在 2005， Gudder 问下列问题：如果一， b 鈭？（ E， 0， 1，鈯？掳) 和鈯 和掳 b 鈯 掳 b，它是盒子那 2 (掳 b ) 鈮 ? 2 鈯 ? b 2？在这份报纸，我们构造一个例子否定地回答这个问题。关键词效果代数学 - 顺序的效果代数学 - 平均价值不平等先生(2000 ) 题目分类 81P15 由中国的自然科学资金支持了(资助 Nos. 10771191 并且 10471124 )