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

Abstract

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-Émery-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. Under the same assumption we also establish that the set of density matrices with finite entropy endowed with the noncommutative transport metric is a geodesic space and that the entropy is semi-convex along these geodesics.

References in corpus (2)

• Gradient flow structure for McKean-Vlasov equations on discrete spaces
• $L^2$-Wasserstein distances of tracial $W^{\ast }$-algebras and their disintegration problem

Cited by in corpus (7)

• A dual formula for the noncommutative transport distance
• Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance
• Ricci curvature of quantum channels on non-commutative transportation metric spaces
• Graph Hörmander Systems
• Geometric Approach Towards Complete Logarithmic Sobolev Inequalities
• Quantum optimal transport for approximately finite-dimensional $C^{\ast }$-algebras
• Curvature-dimension conditions for symmetric quantum Markov semigroups