A Noncommutative Transport Metric and Symmetric Quantum Markov Semigroups as Gradient Flows of the Entropy

We study quantum Dirichlet formsand the associated symmetric quantum Markov semigroupson noncommutative $L^2$ spaces. It is known from the work of Cipriani and Sauvageot that these semigroupsinduce a first order differential calculus, and we use this differential calculusto define a noncommutative transport metric on the set of density matrices. This construction generalizes both the $L^2$-Wasserstein distance on a large class of metric spaces as well as the discrete transport distance introduced by Maas, Mielke, and Chow--Huang--Li--Zhou. Assuming a Bakry--\'Emery-type gradient estimate, we show that the quantum Markov semigroupcan be viewed as a metric gradient flow of the entropy with respect to this transport metric. Under the same assumption we also establish that the set of density matrices with finite entropy endowed with the noncommutative transport metric is a geodesicspace and that the entropy is semi-convex along these geodesics.