Macroscopic Thermodynamic Reversibility in Quantum Many-Body Systems

The resource theory of thermal operations, an established model for small-scale thermodynamics, provides an extension of equilibrium thermodynamics to nonequilibrium situations. On a lattice of any dimension with any translation-invariant local Hamiltonian, we identify a large set of translation-invariant states that can be reversibly converted to and from the thermal state with thermal operations and a small amount of coherence. These are the spatially ergodic states, i.e., states that have sharp statistics for any translation-invariant observable, and mixtures of such states with the same thermodynamic potential. As an intermediate result, we show for a general state that if the gap between the min- and the max-relative entropies to the thermal state is small, then the state can be approximately reversibly converted to and from the thermal state with thermal operations and a small source of coherence. Our proof provides a quantum version of the Shannon-McMillan-Breiman theorem for the relative entropy and a quantum Stein’s lemma for ergodic states and local Gibbs states. Our results provide a strong link between the abstract resource theory of thermodynamics and more realistic physical systems as we achieve a robust and operational characterization of the emergence of a thermodynamic potential in translation-invariant lattice systems.

Figure 1

A thermodynamic potential emerges when the underlying resource theory is reversible. (a) For a state $\overline{\rho }$ that is block-diagonal in energy, the work that can be extracted is given by the min-relative entropy $W_{\mathrm{dist}}={\beta }^{-1}{S}_{\min }^{\epsilon }(\overline{\rho }\parallel \gamma )$, leaving the system in the thermal state $\gamma =e^{-\beta H}/\mathrm{tr}(e^{-\beta H})$. Conversely, the work required to prepare $\overline{\rho }$ from the thermal state is $W_{\mathrm{form}}={\beta }^{-1}{S}_{\max }^{\epsilon }(\overline{\rho }\parallel \gamma )$. (b) Suppose a state $\rho$ (respectively ${\rho }^{\prime }$) can be reversibly converted to and from the thermal state with work $F(\rho )-F(\gamma )$ [respectively, $F({\rho }^{\prime })-F(\gamma )$]. Then $\rho$ and ${\rho }^{\prime }$ can be reversely interconverted. In this case the resource theory is said to be reversible, and the thermodynamic potential $F(\rho )$ fully characterizes the work required for state transformations. (c) As an intermediate result, we show that if the min- and the max-relative entropies of any arbitrary quantum state $\rho$ coincide approximately, then coherences in the state are suppressed, making it nearly block diagonal in energy. The state is then approximately reversibly convertible to and from the thermal state with thermal operations and a small source of coherence.

Figure 2

Ergodic state on a lattice. An ergodic state is one that is translation-invariant and that produces sharp statistics for any translation-invariant observable. Our main result is to show that any two ergodic states can be reversibly interconverted with thermal operations and a sublinear amount of coherence, with the reversible work cost deriving from a thermodynamic potential given by the Kullback-Leibler divergence. Furthermore, a translation-invariant state has a thermodynamic potential if and only if it is a mixture of ergodic states of equal potential, providing a robust and operational understanding of the emergence of a thermodynamic potential in lattice systems.