  For quantum spin systems in any spatial dimension with a local,
translation-invariant Hamiltonian, we prove that asymptotic state
convertibility from a quantum state to another one by a thermodynamically
feasible class of quantum dynamics, called thermal operations, is completely
characterized by the Kullback-Leibler (KL) divergence rate, if the state is
translation-invariant and spatially ergodic. Our proof consists of two parts
and is phrased in terms of a branch of the quantum information theory called
the resource theory. First, we prove that any states, for which the min and max
R\'enyi divergences collapse approximately to a single value, can be
approximately reversibly converted into one another by thermal operations with
the aid of a small source of quantum coherence. Second, we prove that these
divergences collapse asymptotically to the KL divergence rate for any
translation-invariant ergodic state. We show this via a generalization of the
quantum Stein's lemma for quantum hypothesis testing beyond independent and
identically distributed (i.i.d.) situations. Our result implies that the KL
divergence rate serves as a thermodynamic potential that provides a complete
characterization of thermodynamic convertibility of ergodic states of quantum
many-body systems in the thermodynamic limit, including out-of-equilibrium and
fully quantum situations.
