Several recent results on thermodynamics have been obtained using the tools
of quantum information theory and resource theories. So far, the resource
theories utilised to describe thermodynamics have assumed the existence of an
infinite thermal reservoir, by declaring that thermal states at some background
temperature come for free. Here, we propose a resource theory of quantum
thermodynamics without a background temperature, so that no states at all come
for free. We apply this resource theory to the case of many non-interacting
systems, and show that all quantum states are classified by their entropy and
average energy, even arbitrarily far away from equilibrium. This implies that
thermodynamics takes place in a two-dimensional convex set that we call the
energy-entropy diagram. The answers to many resource-theoretic questions about
thermodynamics can be read off from this diagram, such as the efficiency of a
heat engine consisting of finite reservoirs, or the rate of conversion between
two states. This allows us to consider a resource theory which puts work and
heat on an equal footing, and serves as a model for other resource theories.Comment: main text: 12 pages, 5 figure; appendix: 7 page
