Quantum detailed balance conditions and fluctuation relations for thermalizing quantum dynamics

TL;DR

This paper demonstrates that quantum detailed balance conditions imply quantum fluctuation relations in thermalizing quantum systems, but not vice versa, clarifying their relationship in quantum thermodynamics.

Contribution

It establishes a one-way implication from quantum detailed balance to fluctuation relations and provides examples illustrating cases where the converse does not hold.

Findings

Quantum detailed balance implies quantum fluctuation relations in thermalizing dynamics.

Quantum fluctuation relations can exist without detailed balance.

Examples demonstrate the non-equivalence of these conditions in certain cases.

Abstract

Quantum detailed balance conditions and quantum fluctuation relations are two important concepts in the dynamics of open quantum systems: both concern how such systems behave when they thermalize because of interaction with an environment. We prove that for thermalizing quantum dynamics the quantum detailed balance conditions yield the validity of a quantum fluctuation relation (where only forward-time dynamics is considered). This implies that to have such a quantum fluctuation relation (which in turn enables a precise formulation of the second law of thermodynamics for quantum systems) it suffices to fulfill the quantum detailed balance conditions. We, however, show that the converse is not necessarily true; indeed, there are cases of thermalizing dynamics which feature the quantum fluctuation relation without satisfying detailed balance. We illustrate our results with three examples.