双随机矩阵与量子通道

IF 0.3 Q4 MATHEMATICS, APPLIED
Haridas kumar Das, Md Kaisar Ahmed
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引用次数: 0

摘要

摘要本文的主要目的是研究具有多数化和Birkhoff定理的双随机矩阵。推广了双随机矩阵特征值的Perron-Frobenius定理。n × n双随机矩阵的所有可能特征值的区域是正则多边形(n - 1)在复平面上的并集。这个命题是由一个著名的猜想来保证的,这个猜想被称为完美米尔斯基猜想,它对n = 1,2,3,4成立,对n = 5不成立。我们用图形表示了n = 1,2,3,4的Perfect-Mirsky区域的极值特征值,并确定了相应的双随机矩阵。考虑到Rivard-Mashreghi在2007年给出的反例,我们为n = 5的猜想引入一个更一般的反例。随后,我们讨论了与量子通道(qc)相关的不同类型的正映射,并最后引入了一个定理来确定量子通道是否产生双随机矩阵。这个证据是直截了当的,使用了矩阵理论和泛函分析的基本工具。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Doubly stochastic matrices and the quantum channels
Abstract The main object of this paper is to study doubly stochastic matrices with majorization and the Birkhoff theorem. The Perron-Frobenius theorem on eigenvalues is generalized for doubly stochastic matrices. The region of all possible eigenvalues of n-by-n doubly stochastic matrix is the union of regular (n – 1) polygons into the complex plane. This statement is ensured by a famous conjecture known as the Perfect-Mirsky conjecture which is true for n = 1, 2, 3, 4 and untrue for n = 5. We show the extremal eigenvalues of the Perfect-Mirsky regions graphically for n = 1, 2, 3, 4 and identify corresponding doubly stochastic matrices. Bearing in mind the counterexample of Rivard-Mashreghi given in 2007, we introduce a more general counterexample to the conjecture for n = 5. Later, we discuss different types of positive maps relevant to Quantum Channels (QCs) and finally introduce a theorem to determine whether a QCs gives rise to a doubly stochastic matrix or not. This evidence is straightforward and uses the basic tools of matrix theory and functional analysis.
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