A Meta Logarithmic-Sobolev Inequality for Phase-Covariant Gaussian Channels

IF 1.4 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Annales Henri Poincaré Pub Date : 2024-09-09 DOI:10.1007/s00023-024-01487-2
Salman Beigi, Saleh Rahimi-Keshari
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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.

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相变高斯信道的元对数-索博列夫不等式
我们为玻色子量子系统的所有单模相变高斯信道的林德布拉德引入了一个元对数-索波列夫(log-Sobolev)不等式,并证明该不等式在热态下是饱和的。我们证明,我们的不等式提供了一个通用框架,用于推导有关相变高斯信道的信息论结果。具体地说,通过使用热状态的最优性,我们明确地计算出了与量子奥恩斯坦-乌伦贝克半群相关的p-log-Sobolev不等式的最优常数(1\le p\le 2\)。在我们的工作之前,最佳常数只在(p=1)时确定。我们的元对数-索博廖夫不等式还使我们能够为单模情况下的受约束最小输出熵猜想提供另一种证明。具体来说,我们证明了对于任何单模相变高斯信道\(\Phi \),冯-诺依曼熵\(S\big (\Phi (\rho )\big)\)在所有单模状态\(\rho \)上的最小值与给定的\(S(\rho )\)下限是在热状态下实现的。
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来源期刊
Annales Henri Poincaré
Annales Henri Poincaré 物理-物理:粒子与场物理
CiteScore
3.00
自引率
6.70%
发文量
108
审稿时长
6-12 weeks
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.
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