Renormalization group and long-range conditional mutual information in hierarchical models

Abstract

A departure of a mixed quantum state from a local Gibbs description is generally invisible to local observables but can be detected by the conditional mutual information (CMI). Here we study the relationship between the renormalization group (RG) and CMI, and in particular, how RG constrains CMI. We first show that the CMI between nonadjacent regions $A$ and $C$, conditioned on the buffer region $B$, is UV-finite whenever the state admits a locally reversible RG with a fixed on-site Hilbert space dimension. We then study two hierarchical models that have long-range CMI and yet admit a simple RG description. The first model has a divergent Markov length at every temperature $0<T<\infty$ but nevertheless flows to an infinite-temperature product state under RG. The second model satisfies the local Markov condition while violating the global one and is stable against weak noise. At the critical noise strength, the two-point CMI decays only polynomially as a function of the system size, while the two-point mutual information vanishes.