Generalized quantum operations and almost sharp quantum effects
In this paper, we study generalized quantum operations and almost sharp quantum effects, our results generalize and improve some important conclusions in [2] and [3].
math-ph↗
arXiv subjects
Shen Jun
Publications and source records attributed to Shen Jun.
Spectral representation of infimum of bounded quantum observables
In 2006, Gudder introduced a logic order on bounded quantum observable set $S(H)$. In 2007, Pulmannova and Vincekova proved that for each subset $\cal D$ of $S(H)$, the infimum of $\cal D$ exists with respect to this logic order. In this paper, we present the spectral representation for the infimum of $\cal D$.
math-ph↗
Sequential product on standard effect algebra ${\cal E} (H)$
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 method
math-ph↗
The average value inequality in sequential effect algebras
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.
math-ph↗
The n-th root of sequential effect algebras
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 $c$ ($k<n$) ? Recently, we answered the strengthened problem affirmatively.
math-ph↗
Remarks on the sequential effect algebras
In this paper, first, we answer affirmatively an open problem which was presented in 2005 by professor Gudder on the sub-sequential effect algebras. That is, we prove that if $(E,0,1, \oplus, \circ)$ is a sequential effect algebra and $A$ is a commutative subset of $E$, then the sub-sequential effect algebra $\bar{A}$ generated by $A$ is also commutative. Next, we also study the following uniqueness problem: If $na=nb=c$ for some positive integer $n\geq 2$, then under what conditions $a=b$ hold? We prove that if $c$ is a sharp element of $E$ and $a|b$, then $a=b$. We give also two examples to show that neither of the above two conditions can be discarded.
math-ph↗
Not each sequential effect algebra is sharply dominating
Let $E$ be an effect algebra and $E_S$ be the set of all sharp elements of $E$. $E$ is said to be sharply dominating if for each $a\in E$ there exists a smallest element $\widehat{a}\in E_s$ such that $a\leq \widehat{a}$. In 2002, Professors Gudder and Greechie proved that each $\sigma$-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.
math-ph↗