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

• URL: http://arxiv.org/abs/2311.09572v2
• Date: Sat, 28 Sep 2024 04:27:46 GMT
• Title: A Meta Logarithmic-Sobolev Inequality for Phase-Covariant Gaussian Channels
• Authors: Salman Beigi, Saleh Rahimi-Keshari,
• Abstract summary: We show that our inequality provides a general framework to derive information theoretic results regarding phase-covariant Gaussian channels. Specifically, we explicitly compute the optimal constant $alpha_p$, for $1leq pleq 2$, of the $p$-log-Sobolev inequality associated to the quantum Ornstein-Uhlenbeck semigroup.
• Score: 3.130722489512822
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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 $1\leq p\leq 2$, 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 $S\big(\Phi(\rho)\big)$ over all single-mode states $\rho$ with a given lower bound on $S(\rho)$, is achieved at a thermal state.

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