{"title":"Zero-error correctibility and phase retrievability for twirling channels","authors":"Deguang Han, 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":null,"pages":null},"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
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
. 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
. We also present a lower bound for the phase retrievability in terms of the minimal length of phase retrievable frames for ℂn
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.