  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. 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.
