Dependence of integrated, instantaneous, and fluctuating entropy production on the initial state in quantum and classical processes

TL;DR

This paper establishes a universal information-theoretic framework linking entropy production in quantum and classical processes to initial state mismatch, providing new bounds and operational measures for thermodynamic irreversibility.

Contribution

It introduces a universal form for the mismatch cost of initial states on entropy production, applicable to integrated, fluctuating, and instantaneous EP, and connects thermodynamic irreversibility with logical irreversibility.

Findings

Mismatch cost equals the contraction of relative entropy over time.

Fluctuating EP mismatch cost obeys an integral fluctuation theorem.

Results apply to both finite and infinite dimensional systems.

Abstract

We consider the additional entropy production (EP) incurred by a fixed quantum or classical process on some initial state $\rho$, above the minimum EP incurred by the same process on any initial state. We show that this additional EP, which we term the "mismatch cost of $\rho$", has a universal information-theoretic form: it is given by the contraction of the relative entropy between $\rho$ and the least-dissipative initial state $\varphi$ over time. We derive versions of this result for integrated EP incurred over the course of a process, for trajectory-level fluctuating EP, and for instantaneous EP rate. We also show that mismatch cost for fluctuating EP obeys an integral fluctuation theorem. Our results demonstrate a fundamental relationship between "thermodynamic irreversibility" (generation of EP) and "logical irreversibility" (inability to know the initial state corresponding to a given final state). We use this relationship to derive quantitative bounds on the thermodynamics of quantum error correction and to propose a thermodynamically-operationalized measure of the logical irreversibility of a quantum channel. Our results hold for both finite and infinite dimensional systems, and generalize beyond EP to many other thermodynamic costs, including nonadiabatic EP, free energy loss, and entropy gain.