Doubly stochastic matrices and the quantum channels

IF 0.3 Q4 MATHEMATICS, APPLIED
Haridas kumar Das, Md Kaisar Ahmed
{"title":"Doubly stochastic matrices and the quantum channels","authors":"Haridas kumar Das, Md Kaisar Ahmed","doi":"10.2478/jamsi-2021-0005","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":43016,"journal":{"name":"Journal of Applied Mathematics Statistics and Informatics","volume":"17 1","pages":"73 - 107"},"PeriodicalIF":0.3000,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied Mathematics Statistics and Informatics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/jamsi-2021-0005","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0

Abstract

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.
双随机矩阵与量子通道
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。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
8
审稿时长
20 weeks
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信