Rapid mixing implies exponential decay of correlations

TL;DR

This paper demonstrates that a spectral gap or Log-Sobolev constant in open quantum systems ensures exponential decay of correlations, leading to area laws and stability, with implications for quantum simulation.

Contribution

It establishes a link between spectral gap, Log-Sobolev constant, and correlation decay in open quantum systems, including fermionic systems, with new stability and simulation insights.

Findings

Spectral gap implies exponential decay of correlations.

Log-Sobolev constant ensures clustering of mutual information.

Gapped free-fermionic systems exhibit correlation clustering.

Abstract

We provide an analysis of the correlation properties of spin and fermionic systems on a lattice evolving according to open system dynamics generated by a local primitive Liouvillian. We show that if the Liouvillian has a spectral gap which is independent of the system size, then the correlations between local observables decay exponentially as a function of the distance between their supports. We prove, furthermore, that if the Log-Sobolev constant is independent of the system size, then the system satisfies clustering of correlations in the mutual information - a much more stringent form of correlation decay. As a consequence, in the latter case we get an area law (with logarithmic corrections) for the mutual information. As a further corollary, we obtain a stability theorem for local distant perturbations. We also demonstrate that gapped free-fermionic systems exhibit clustering of correlations in the covariance and in the mutual information. We conclude with a discussion of the implications of these results for the classical simulation of open quantum systems with matrix-product operators and the robust dissipative preparation of topologically ordered states of lattice spin systems.