Quantum Entropy Contraction and Factorization from Hypercontractivity

Abstract: We prove that hypercontractivity implies entropy contraction for a single quantum channel, without a detailed balance condition. For primitive quantum Markov semigroups that are KMS-symmetric with respect to a faithful invariant state (σ), we obtain the modified Log-Sobolev bound $$α<em>1\geq \fracλ{(2+\log|σ^{-1}|</em>\infty)}$$ where $λ$ is the spectral gap. This removes the assumption of (L_p)-regularity for the comparison through the log-Sobolev constant. As an application, we show that the hypercontractivity of an average of two conditional expectations implies the approximate tensorization of relative entropy.