Zero-error correctibility and phase retrievability for twirling channels

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL Reports on Mathematical Physics Pub Date : 2024-02-01 DOI:10.1016/S0034-4877(24)00012-0
Deguang Han, Kai Liu
{"title":"Zero-error correctibility and phase retrievability for twirling channels","authors":"Deguang Han,&nbsp;Kai Liu","doi":"10.1016/S0034-4877(24)00012-0","DOIUrl":null,"url":null,"abstract":"<div><p>A twirling channel is a quantum channel induced by a continuous unitary representation\n<span><math><mrow><mi>π</mi><mo>=</mo><msubsup><mo>∑</mo><mi>i</mi><mo>⊕</mo></msubsup><mrow><msub><mi>m</mi><mi>i</mi></msub><msub><mi>π</mi><mi>i</mi></msub></mrow></mrow></math></span> on a compact group <em>G</em>, where π<sub><em>i</em></sub> are inequivalent irreducible representations. Motivated by a recent work [<span>8</span>] on minimal mixed unitary rank of Φ<em><sub>π</sub></em>, we explore the connections of the independence number, zero-error capacity, quantum codes, orthogonality index and phase retrievability of the quantum channel Φ<em><sub>π</sub></em> with the irreducible representation multiplicities <em>m<sub>i</sub></em> and the irreducible representation dimensions dim\n<span><math><mrow><msub><mi>H</mi><mrow><msub><mi>π</mi><mi>i</mi></msub></mrow></msub></mrow></math></span>. In particular, we show that the independence number of Φ<em><sub>π</sub></em> is the sum of the multiplicities, the orthogonal index of Φ<em><sub>π</sub></em> is exactly the sum of those representation dimensions, and the zero-error capacity is equal to log\n<span><math><mrow><mrow><mo>(</mo><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></msubsup><mrow><msub><mi>m</mi><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow></math></span>. We also present a lower bound for the phase retrievability in terms of the minimal length of phase retrievable frames for ℂ<sup><em>n</em></sup></p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"93 1","pages":"Pages 87-102"},"PeriodicalIF":1.0000,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0034487724000120/pdfft?md5=c13c155f18d2d613dd2d69aa31d00367&pid=1-s2.0-S0034487724000120-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reports on Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0034487724000120","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0

Abstract

A twirling channel is a quantum channel induced by a continuous unitary representation π=imiπi on a compact group G, where πi are inequivalent irreducible representations. Motivated by a recent work [8] on minimal mixed unitary rank of Φπ, we explore the connections of the independence number, zero-error capacity, quantum codes, orthogonality index and phase retrievability of the quantum channel Φπ with the irreducible representation multiplicities mi and the irreducible representation dimensions dim Hπi. In particular, we show that the independence number of Φπ is the sum of the multiplicities, the orthogonal index of Φπ is exactly the sum of those representation dimensions, and the zero-error capacity is equal to log (i=1dmi). We also present a lower bound for the phase retrievability in terms of the minimal length of phase retrievable frames for ℂn

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
旋转信道的零误差可纠正性和相位可检索性
扭转信道是由紧凑群 G 上的连续单元表示π=∑i⊕miπi 所诱导的量子信道,其中 πi 是不等价的不可还原表示。受最近关于 Φπ 的最小混合单元秩的研究[8]的启发,我们探讨了量子信道 Φπ 的独立数、零错误容量、量子密码、正交指数和相位可检索性与不可还原表征乘数 mi 和不可还原表征维数 dimHπi 之间的联系。我们特别指出,Φπ 的独立数是乘数之和,Φπ 的正交索引正好是这些表示维数之和,零误差容量等于 log(∑i=1dmi)。我们还根据ℂn 的相位可检索帧的最小长度提出了相位可检索性下限。
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 去求助
来源期刊
Reports on Mathematical Physics
Reports on Mathematical Physics 物理-物理:数学物理
CiteScore
1.80
自引率
0.00%
发文量
40
审稿时长
6 months
期刊介绍: Reports on Mathematical Physics publish papers in theoretical physics which present a rigorous mathematical approach to problems of quantum and classical mechanics and field theories, relativity and gravitation, statistical physics, thermodynamics, mathematical foundations of physical theories, etc. Preferred are papers using modern methods of functional analysis, probability theory, differential geometry, algebra and mathematical logic. Papers without direct connection with physics will not be accepted. Manuscripts should be concise, but possibly complete in presentation and discussion, to be comprehensible not only for mathematicians, but also for mathematically oriented theoretical physicists. All papers should describe original work and be written in English.
期刊最新文献
New operator realization of angular momentum for description of electron's motion in uniform magnetic field Construction of quadratic invariants for time-dependent systems in complex phase space using Struckmeier and Riedel approach Higher-order squeezing of both quadrature components in superposition of orthogonal even coherent state and vacuum state Weakly periodic gibbs measures for the HC model with a countable set of spin values Certain advancements in multidimensional q-hermite polynomials
×
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
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 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 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1