Fundamental work costs of preparation and erasure in the presence of quantum side information

The thought experiment of Maxwell's demon highlights the effect of side information in thermodynamics. In this paper, we present an axiomatic treatment of a quantum Maxwell's demon, by introducing a resource-theoretic framework of quantum thermodynamics in the presence of quantum side information. Under minimal operational assumptions that capture the demon's behaviour, we derive the one-shot work costs of preparing, as well as erasing, a thermodynamic system whose coupling with the demon's mind is described by a bipartite quantum state. With trivial Hamiltonians, these work costs are precisely captured by the smoothed conditional min- and max-entropies, respectively, thus providing operational interpretations for these one-shot information-theoretic quantities in microscopic thermodynamics. An immediate, information-theoretic implication of our results is an affirmative proof of the conjectured maximality of the conditional max-entropy among all axiomatically plausible conditional entropies, complementing the recently established minimality of the conditional min-entropy. We then generalize our main results to the setting with nontrivial Hamiltonians, wherein the work costs of preparation and erasure are captured by a generalized type of mutual information. Finally, we present a macroscopic second law of thermodynamics in the presence of quantum side information, in terms of a conditional version of the Helmholtz free energy. Our results extend the conceptual connection between thermodynamics and quantum information theory by refining the axiomatic common ground between the two theories and revealing fundamental insights of each theory in light of the other.