Conditional Independence of 1D Gibbs States with Applications to Efficient Learning

Understand

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.