A Wasserstein-type Distance to Measure Deviation from Equilibrium of Quantum Markov Semigroups

Abstract

In this paper we define a distance W between states in the non-commutative von Neumann algebra [Formula: see text] of bounded operators on a separable Hilbert space [Formula: see text], in order to measure deviations from equilibrium using a rate ep W(·). The restriction of W to the diagonal subalgebra of [Formula: see text] coincides with the Wasserstein distance used in optimal transport. Moreover, if ρ is a normal invariant state of a quantum Markov semigroup [Formula: see text], then ep W(ρ) = 0 if and only if a detailed balance condition holds.