Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems

Abstract

We study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $C^{\ast }$-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.

An error in Theorem 10.6 has been corrected

References in corpus (1)

• An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models

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