Entropy Production for Quantum Markov Semigroups

TL;DR

This paper introduces a quantitative measure called entropy production for quantum Markov semigroups, which indicates how far a state is from equilibrium, and provides an explicit formula linking it to the generator's properties.

Contribution

It defines a novel entropy production index for quantum Markov semigroups and establishes its relation to quantum detailed balance conditions.

Findings

Entropy production is zero iff the quantum detailed balance condition holds.

Explicit trace formula for entropy production in terms of the generator.

Entropy production quantifies deviation from equilibrium in quantum systems.

Abstract

An invariant state of a quantum Markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition. In this paper, we introduce a notion of entropy production for faithful normal invariant states of a quantum Markov semigroup on B(h) as a numerical index measuring "how much far" they are from equilibrium. The entropy production is defined as the derivative of the relative entropy of the one-step forward and backward evolution in analogy with the classical probabilistic concept. We prove an explicit trace formula expressing the entropy production in terms of the completely positive part of the generator of a norm continuous quantum Markov semigroup showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds.