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 α_p, for 1≤ p≤ 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 Φ, the minimum of the von Neumann entropy S(Φ(ρ)) over all single-mode states ρ with a given lower bound on S(ρ), is achieved at a thermal state.