  We show that spin chains in thermal equilibrium have a correlation structure
in which individual regions are strongly correlated at most with their near
vicinity. We quantify this with alternative notions of the conditional mutual
information defined through the so-called Belavkin-Staszewski relative entropy.
We prove that these measures decay superexponentially at every positive
temperature, under the assumption that the spin chain Hamiltonian is
translation-invariant. Using a recovery map associated with these measures, we
sequentially construct tensor network approximations in terms of marginals of
small (sublogarithmic) size. As a main application, we show that classical
representations of the states can be learned efficiently from local
measurements with a polynomial sample complexity. We also prove an approximate
factorization condition for the purity of the entire Gibbs state, which implies
that it can be efficiently estimated to a small multiplicative error from a
small number of local measurements. The results extend from strictly local to
exponentially-decaying interactions above a threshold temperature, albeit only
with exponential decay rates. As a technical step of independent interest, we
show an upper bound to the decay of the Belavkin-Staszewski relative entropy
upon the application of a conditional expectation.
