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

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 to 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 $S(\Phi (\rho ))$ over all single-mode states $\rho$ with a given lower bound on $S(\rho )$, is achieved at a thermal state.

38 pages

References in corpus (21)

• Quantum information with continuous variables
• Gaussian Quantum Information
• Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
• Minimum output entropy of bosonic channels: a conjecture
• Quantum logarithmic Sobolev inequalities and rapid mixing
• A solution of the Gaussian optimizer conjecture
• Classical capacity of bosonic broadcast communication and a new minimum output entropy conjecture
• A generalization of the Entropy Power Inequality to Bosonic Quantum Systems
• Complete entropic inequalities for quantum Markov chains
• Majorization and additivity for multimode bosonic Gaussian channels
• Gaussian optimizers and the additivity problem in quantum information theory
• Passive states optimize the output of bosonic Gaussian quantum channels
• Generalized minimal output entropy conjecture for one-mode Gaussian channels: definitions and some exact results
• Gaussian States Minimize the Output Entropy of the One-Mode Quantum Attenuator
• Gaussian states minimize the output entropy of one-mode quantum Gaussian channels
• Gaussian optimizers for entropic inequalities in quantum information
• Quantum reverse hypercontractivity
• The conditional entropy power inequality for quantum additive noise channels
• Geometric inequalities from phase space translations
• The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have Gaussian Maximizers
• New lower bounds to the output entropy of multi-mode quantum Gaussian channels

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