  The ideas of thermodynamics have proved fruitful in the setting of quantum
information theory, in particular the notion that when the allowed
transformations of a system are restricted, certain states of the system become
useful resources with which one can prepare previously inaccessible states. The
theory of entanglement is perhaps the best-known and most well-understood
resource theory in this sense. Here we return to the basic questions of
thermodynamics using the formalism of resource theories developed in quantum
information theory and show that the free energy of thermodynamics emerges
naturally from the resource theory of energy-preserving transformations.
Specifically, the free energy quantifies the amount of useful work which can be
extracted from asymptotically-many copies of a quantum system when using only
reversible energy-preserving transformations and a thermal bath at fixed
temperature. The free energy also quantifies the rate at which resource states
can be reversibly interconverted asymptotically, provided that a sublinear
amount of coherent superposition over energy levels is available, a situation
analogous to the sublinear amount of classical communication required for
entanglement dilution.
