Thermodynamic Capacity of Quantum Processes

Abstract

Thermodynamics imposes restrictions on what state transformations are possible. In the macroscopic limit of asymptotically many independent copies of a state-as for instance in the case of an ideal gas-the possible transformations become reversible and are fully characterized by the free energy. In this letter, we present a thermodynamic resource theory for quantum processes that also becomes reversible in the macroscopic limit. Namely, we identify a unique single-letter and additive quantity, the thermodynamic capacity, that characterizes the "thermodynamic value" of a quantum channel. As a consequence the work required to simulate many repetitions of a quantum process employing many repetitions of another quantum process becomes equal to the difference of the respective thermodynamic capacities. For our proof, we construct explicit universal implementations of quantum processes using Gibbs-preserving maps and a battery, requiring an amount of work asymptotically equal to the thermodynamic capacity. This implementation is also possible with thermal operations in the case of time-covariant quantum processes or when restricting to independent and identical inputs. In our derivations we make extensive use of Schur-Weyl duality and other information-theoretic tools, leading to a generalized notion of quantum typical subspaces. APPENDICESThe appendices are structured as follows. Appendix A introduces the necessary preliminaries and fixes some notation. In Appendix B we introduce the thermodynamic capacity and calculate the thermodynamic capacity of some simple example channels. Appendix C then reviews some additional tools that we need based on Schur-Weyl duality, such as universal entropy and energy expectation value estimation, as well as the postselection technique. In Appendix D, we consider the case of systems described by a trivial Hamiltonian, and we prove our main result in this situation by using a reasoning similar to refs. [32,49]. We also show that this reasoning fails in the general case, because the coherent relative entropy does not

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