Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

Eric A. Carlen, Jan Maas

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional C-algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

Original languageEnglish (US)
Pages (from-to)319-378
Number of pages60
JournalJournal of Statistical Physics
Volume178
Issue number2
DOIs
StatePublished - Jan 1 2020

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Keywords

  • Functional inequalities
  • Gradient flow
  • Lindblad equation
  • Non-commutative optimal transport

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