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arXiv:0903.5120

The n-th root of sequential effect algebras

Jun Shen, Junde Wu

Published Mar 30, 2009• 0 citations
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Abstract

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 $n>1$, is there a sequential effect algebra such that the n-th root of its some element $c$ is not unique and the n-th root of $c$ is not the k-th root of $c$ ($k

DOI
10.48550/arxiv.0903.5120

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The nth root of sequential effect algebras

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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.
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
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.