The connections between renormalization in statistical mechanics and information theory are intuitively evident, but a satisfactory theoretical treatment remains elusive. We show that the real space renormalization map that minimizes long range couplings in the renormalized Hamiltonian is, somewhat counterintuitively, the one that minimizes the loss of short-range mutual information between a block and its boundary. Moreover, we show that a previously proposed minimization focusing on preserving long-range mutual information is a relaxation of this approach, which indicates that the aims of preserving long-range physics and eliminating short-range couplings are related in a nontrivial way.
