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

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We study quantum Dirichlet forms and the associated symmetric quantum Markov semigroups on noncommutative $L^2$ spaces.

• It is known from the work of Cipriani and Sauvageot that these semigroups induce a first order differential calculus, and we use this differential calculus to 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 semigroup can be viewed as a metric gradient flow of the entropy with respect to this transport metric.