{"title":"A Meta Logarithmic-Sobolev Inequality for Phase-Covariant Gaussian Channels","authors":"Salman Beigi, Saleh Rahimi-Keshari","doi":"10.1007/s00023-024-01487-2","DOIUrl":null,"url":null,"abstract":"<p>We introduce a meta logarithmic-Sobolev (log-Sobolev) inequality for the Lindbladian of all single-mode phase-covariant Gaussian channels of bosonic quantum systems and prove that this inequality is saturated by thermal states. We show that our inequality provides a general framework to derive information theoretic results regarding phase-covariant Gaussian channels. Specifically, by using the optimality of thermal states, we explicitly compute the optimal constant <span>\\(\\alpha _p\\)</span>, for <span>\\(1\\le p\\le 2\\)</span>, of the <i>p</i>-log-Sobolev inequality associated with the quantum Ornstein–Uhlenbeck semigroup. Prior to our work, the optimal constant was only determined for <span>\\(p=1\\)</span>. Our meta log-Sobolev inequality also enables us to provide an alternative proof for the constrained minimum output entropy conjecture in the single-mode case. Specifically, we show that for any single-mode phase-covariant Gaussian channel <span>\\(\\Phi \\)</span>, the minimum of the von Neumann entropy <span>\\(S\\big (\\Phi (\\rho )\\big )\\)</span> over all single-mode states <span>\\(\\rho \\)</span> with a given lower bound on <span>\\(S(\\rho )\\)</span> is achieved at a thermal state.\n</p>","PeriodicalId":463,"journal":{"name":"Annales Henri Poincaré","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales Henri Poincaré","FirstCategoryId":"4","ListUrlMain":"https://doi.org/10.1007/s00023-024-01487-2","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce a meta logarithmic-Sobolev (log-Sobolev) inequality for the Lindbladian of all single-mode phase-covariant Gaussian channels of bosonic quantum systems and prove that this inequality is saturated by thermal states. We show that our inequality provides a general framework to derive information theoretic results regarding phase-covariant Gaussian channels. Specifically, by using the optimality of thermal states, we explicitly compute the optimal constant \(\alpha _p\), for \(1\le p\le 2\), of the p-log-Sobolev inequality associated with the quantum Ornstein–Uhlenbeck semigroup. Prior to our work, the optimal constant was only determined for \(p=1\). Our meta log-Sobolev inequality also enables us to provide an alternative proof for the constrained minimum output entropy conjecture in the single-mode case. Specifically, we show that for any single-mode phase-covariant Gaussian channel \(\Phi \), the minimum of the von Neumann entropy \(S\big (\Phi (\rho )\big )\) over all single-mode states \(\rho \) with a given lower bound on \(S(\rho )\) is achieved at a thermal state.
期刊介绍:
The two journals Annales de l''Institut Henri Poincaré, physique théorique and Helvetica Physical Acta merged into a single new journal under the name Annales Henri Poincaré - A Journal of Theoretical and Mathematical Physics edited jointly by the Institut Henri Poincaré and by the Swiss Physical Society.
The goal of the journal is to serve the international scientific community in theoretical and mathematical physics by collecting and publishing original research papers meeting the highest professional standards in the field. The emphasis will be on analytical theoretical and mathematical physics in a broad sense.