Quantum Entropy Contraction and Factorization from Hypercontractivity

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 $α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}$ 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.

Source: arXiv:2609.20753v1 - http://arxiv.org/abs/2609.20753v1 PDF: https://arxiv.org/pdf/2609.20753v1 Original Link: http://arxiv.org/abs/2609.20753v1