Strong decay of correlations for Gibbs states in any dimension

Abstract

Quantum systems in thermal equilibrium are described using Gibbs states. The correlations in such states determine how difficult it is to describe or simulate them. In this article, we show that systems with short-range interactions that are above a critical temperature satisfy a mixing condition, that is that for any regions \(A\), \(C\) the distance of the reduced state \(\rho_{AC}\) on these regions to the product of its marginals, \[\| \rho_{AC} \rho_A^{-1} \otimes \rho_C^{-1} - \mathbf{1}_{AC}\| \, ,\] decays exponentially with the distance between regions \(A\) and \(C\). This mixing condition is stronger than other commonly studied measures of correlation. In particular, it implies the exponential decay of the mutual information between distant regions. The mixing condition has been used, for example, to prove positive log-Sobolev constants. On the way, we investigate the relations to other notions of decay of correlations in quantum many-body systems and show that many of them are equivalent under the assumption that there exists a local effective Hamiltonian. The proof employs a variety of tools such as Araki's expansionals and quantum belief propagation.