  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.
