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

We consider the additional entropy production (EP) incurred by a fixed quantum or classical process on some initial state $\rho$, above the minimal EP incurred by the same process on the least-dissipative initial state $\varphi$. 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 $\varphi$ 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.