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Quantum systems in thermal equilibrium are described using Gibbs states. Thecorrelations in such states determine how difficult it is to describe orsimulate them. In this article, we show that systems with short-rangeinteractions 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 exponentiallywith the distance between regions $A$ and $C$. This mixing condition isstronger than other commonly studied measures of correlation. In particular, itimplies the exponential decay of the mutual information between distantregions. The mixing condition has been used, for example, to prove positivelog-Sobolev constants. On the way, we investigate the relations to othernotions of decay of correlations in quantum many-body systems and show thatmany of them are equivalent under the assumption that there exists a localeffective Hamiltonian. The proof employs a variety of tools such as Araki'sexpansionals and quantum belief propagation.

Strong decay of correlations for Gibbs states in any dimension