Mutual information area laws for thermal free fermions

TL;DR

This paper derives an exact asymptotic expression for the mutual information in translationally invariant free fermionic lattice systems, revealing an area law and temperature scaling, with implications for quantum noise and Toeplitz determinant theory.

Contribution

It introduces a novel framework for computing determinants of Toeplitz operators with smooth symbols, enabling precise analysis of mutual information in free fermionic systems.

Findings

Mutual information satisfies an area law with explicit prefactor.

Temperature dependence of mutual information scales logarithmically with inverse temperature.

Provides detailed error bounds and applicability to open quantum systems.

Abstract

We provide a rigorous and asymptotically exact expression of the mutual information of translationally invariant free fermionic lattice systems in a Gibbs state. In order to arrive at this result, we introduce a novel frameworkfor computing determinants of Toeplitz operators with smooth symbols, and for treating Toeplitz matrices with system size dependent entries. The asymptotically exact mutual information for a partition of the one-dimensional lattice satisfies an area law, with a prefactor which we compute explicitly. As examples, we discuss the fermionic XX model in one dimension and free fermionic models on the torus in higher dimensions in detail. Special emphasis is put onto the discussion of the temperature dependence of the mutual information, scaling like the logarithm of the inverse temperature, hence confirming an expression suggested by conformal field theory. We also comment on the applicability of the formalism to treat open systems driven by quantum noise. In the appendix, we derive useful bounds to the mutual information in terms of purities. Finally, we provide a detailed error analysis for finite system sizes. This analysis is valuable in its own right for the abstract theory of Toeplitz determinants.