CMI decay at high temperatures

Let $A$, $B$, and $C$ be regions of a quantum system, and let $\rho_\beta$ be a quantum Gibbs state. Suppose

$$A\cup B\cup C\subseteq\Lambda.$$

The conditional mutual information is

$$I_{\rho_\beta}(A:C|B)=S_{\rho_\beta}(AB)+S_{\rho_\beta}(BC)-S_{\rho_\beta}(ABC)-S_{\rho_\beta}(B),$$

where $S_{\rho_\beta}(X)$ denotes the von Neumann entropy of the reduced state on $X$. CMI decay at high temperatures. For general quantum Gibbs states at sufficiently high temperatures, $I_{\rho_\beta}(A:C|B)$ decays rapidly with the distance between the regions $A$ and $C$. The paper states that this high-temperature claim is resolved for the case $A\cup B\cup C=\Lambda$ using its belief-propagation-channel construction; the subsystem case remains the setting where cluster-expansion difficulties arise.