A quantum entropy production operator

Abstract

We introduce a fully quantum notion of entropy production based on the noncommutative extension of the classical log-ratio between forward and reverse processes. Given a pair of quantum objects associated with the forward and reverse descriptions, we define a Hermitian entropy-production operator whose expectation value is non-negative and equal to the Belavkin--Staszewski relative entropy. The operator satisfies exact integral and detailed fluctuation theorems without requiring commutativity. We then specialize this construction to the case in which the forward process is described by a single quantum channel and the reverse process is defined inferentially, through Bayesian retrodiction relative to a prior state, with the Petz transpose map as the Bayesian inverse. In this setting, the relevant quantities can be evaluated explicitly, leading to a number of natural structural and physical properties. The framework recovers the classical formula in the commutative limit, yields explicit expressions for the average entropy production, and clarifies where the fully quantum case departs from standard thermodynamic expectations.