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

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.