Exponential decay of mutual information for Gibbs states of local Hamiltonians

TL;DR

This paper proves that in 1D quantum spin systems, Gibbs states exhibit exponential decay of correlations and mutual information at any temperature, simplifying their description and analysis.

Contribution

It establishes the equivalence of exponential decay of correlations, clustering, and mutual information decay for 1D Gibbs states at any temperature, extending Araki's results.

Findings

Mutual information decays exponentially with distance in 1D Gibbs states.

Gibbs states satisfy uniform exponential decay of correlations.

Gibbs states nearly saturate the data-processing inequality for Belavkin-Staszewski entropy.

Abstract

The thermal equilibrium properties of physical systems can be described using Gibbs states. It is therefore of great interest to know when such states allow for an easy description. In particular, this is the case if correlations between distant regions are small. In this work, we consider 1D quantum spin systems with local, finite-range, translation-invariant interactions at any temperature. In this setting, we show that Gibbs states satisfy uniform exponential decay of correlations and, moreover, the mutual information between two regions decays exponentially with their distance, irrespective of the temperature. In order to prove the latter, we show that exponential decay of correlations of the infinite-chain thermal states, exponential uniform clustering and exponential decay of the mutual information are equivalent for 1D quantum spin systems with local, finite-range interactions at any temperature. In particular, Araki's seminal results yields that the three conditions hold in the translation-invariant case. The methods we use are based on the Belavkin-Staszewski relative entropy and on techniques developed by Araki. Moreover, we find that the Gibbs states of the systems we consider are superexponentially close to saturating the data-processing inequality for the Belavkin-Staszewski relative entropy.