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

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