One central question in the study of quantum Markov semigroups is to quantify this return to equilibrium. If one uses the entropy as a measure of the deviation from the equilibrium state, this question is closely related to logarithmic Sobolev inequalities. In the classical case, Bakry-Émery theory or optimal transport methods allow to deduce such logarithmic Sobolev inequalities from lower bounds on the Ricci curvature. In this talk I will review a notion of lower Ricci curvature bounds via a gradient estimate that allows to transfer the optimal transport approach to the quantum setting. I will discuss some of its stability properties and show how to obtain lower Ricci curvature bounds for a couple of examples such as quantum tori, free group factors and q-Gaussian algebras. (This talk is based on joint work with Haonan Zhang.)
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