Clustering of Conditional Mutual Information via Quantum Belief-Propagation Channels

Abstract

Conditional mutual information (CMI) has recently attracted significant attention as a key quantity for characterizing quantum correlations in many-body systems. While it is conjectured that CMI decays rapidly in finite-temperature Gibbs states, a complete and general proof remains elusive. In this work, we introduce a new formulation of the problem based on the \emph{belief propagation (BP) channel}, namely a completely positive trace-preserving (CPTP) map that realizes local perturbations of the Hamiltonian. Within this framework, we prove that establishing the quasi-locality of BP channels implies the decay of CMI, thereby reducing the original conjecture to a more tractable problem. We show that such quasi-local BP channels can be constructed under natural physical conditions, such as uniform rapid mixing or uniform clustering. Under these assumptions, we obtain conditional proofs of CMI decay valid at all temperatures. Moreover, because these assumptions are automatically satisfied at high temperatures, our results in that regime yield unconditional proofs of CMI decay. At the same time, in order to better understand the high-temperature behavior of Gibbs states, we revisit the cluster expansion method. Contrary to common intuition, we demonstrate that when multipartite correlations such as CMI are considered, the cluster expansion suffers from intrinsic divergence problems rooted in the Baker--Campbell--Hausdorff formula, revealing fundamental limitations of this traditional approach.