  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 if the Gibbs state of a quantum
system satisfies that each of its marginals admits a local effective
Hamiltonian with short-range interactions, then it satisfies a mixing
condition, that is, for any regions $A$, $C$ the distance of the reduced state
$\rho_{AC}$ on these regions to the product of its marginals,
  $$\| \rho_{AC} \rho_A^{-1} \otimes \rho_C^{-1} - 1_{AC}\|\, ,$$
  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 prove that the
the condition regarding local effective Hamiltonian is satisfied if the
Hamiltonian of the system is commuting and also commutes with every marginal of
the Gibbs state. The proof of these results employs a variety of tools such as
Araki's expansionals, quantum belief propagation and cluster expansions.
