# Asymptotic Reversibility of Thermal Operations for Interacting Quantum Spin Systems via Generalized Quantum Stein's Lemma

## TL;DR

This paper demonstrates that for translation-invariant ergodic quantum spin systems, the asymptotic convertibility of states under thermal operations is fully characterized by the KL divergence rate, extending quantum Stein's lemma.

## Contribution

It generalizes quantum Stein's lemma to non-i.i.d. ergodic states, establishing KL divergence rate as a thermodynamic potential for quantum many-body systems.

## Key findings

- KL divergence rate characterizes state convertibility
- Extension of quantum Stein's lemma beyond i.i.d. cases
- Reversible conversion of states with quantum coherence

## Abstract

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.