Mutual entropy and thermal area law in C*-algebraic quantum lattice systems
Hajime Moriya
TL;DR
This work develops a rigorous $C^{\ast}$-algebraic framework for quantum mutual entropy in infinitely extended lattice systems and proves a thermal area law via local thermodynamical stability, expressed through surface energies $H_{\partial A}$ and the $v.H.$ limit. It shows that for translation-invariant finite-range 1D spin and fermion lattices at any $\beta>0$, the mutual entropy between left and right halves is finite, bounded by $I_{\varphi}(\mathbb Z_{\mathrm L}:\mathbb Z_{\mathrm R}) \le 2\beta\|W_{\mathrm L,\mathrm R}\|$, and further demonstrates thermal destruction of long-range entanglement as temperature increases. The approach leverages Araki-Gibbs product structures, Donald's formula for quantum mutual entropy, and perturbation bounds to relate global equilibrium states to decoupled local states, without invoking box constructions. The results bridge finite- and infinite-volume statistical mechanics, offer a path toward thermal area laws in AQFT, and illuminate how temperature suppresses quantum correlations in critical regimes. Collectively, the paper provides a principled, model-agnostic method to quantify and bound correlations in infinite quantum lattices and sets the stage for extensions to continuous quantum systems and modular-hamiltonian analyses.
Abstract
We present a general definition of quantum mutual entropy for infinitely extended quantum spin and fermion lattice systems. Using this, we establish a thermal area law in these infinitely extended quantum systems. The proof is based on the local thermodynamical stability (LTS), a variational principle in terms of the conditional free energy. Our thermal area law in quasi-local C*-systems applies to general interactions with well-defined surface energies. We also examine the quantum mutual entropy between the left- and right-sided infinite regions of one-dimensional lattice systems. For general translation-invariant finite-range interactions on such systems, the thermal equilibrium state at any temperature exhibits a finite mutual entropy between these infinite disjoint regions. This further implies that the infinitely large quantum entanglement characteristic of critical ground states in one-dimensional systems is drastically destroyed by even a small positive temperature.