English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study quantum Dirichlet forms and the associated symmetric quantum Markovsemigroups on noncommutative $L^2$ spaces. It is known from the work ofCipriani and Sauvageot that these semigroups induce a first order differentialcalculus, and we use this differential calculus to define a noncommutativetransport metric on the set of density matrices. This construction generalizesboth the $L^2$-Wasserstein distance on a large class of metric spaces as wellas the discrete transport distance introduced by Maas, Mielke, andChow-Huang-Li-Zhou. Assuming a Bakry-\'Emery-type gradient estimate, we showthat the quantum Markov semigroup can be viewed as a metric gradient flow ofthe entropy with respect to this transport metric. Under the same assumption wealso establish that the set of density matrices with finite entropy endowedwith the noncommutative transport metric is a geodesic space and that theentropy is semi-convex along these geodesics.

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