We prove that for a symmetric Markov semigroup, Ricci curvature bounded from below by a non-positive constant combined with a finite L∞-mixing time implies the modified log-Sobolev inequality. Such L∞-mixing time estimates always hold for Markov semigroups that have spectral gap and finite Varopoulos dimension. Our results apply to non-ergodic quantum Markov semigroups with noncommutative Ricci curvature bounds recently introduced by Carlen and Maas. As an application, we prove that the heat semigroup on a compact Riemannian manifold admits a uniform modified log-Sobolev inequality for all its matrix-valued extensions.
- Heat semigroup
- Logarithmic Sobolev inequality
- Quantum Markov semigroup
- Ricci curvature
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