CMI decay at arbitrary temperatures

Let $A$, $B$, and $C$ be regions of a quantum system $\rho_\beta$, with

$$A\cup B\cup C=\Lambda,$$

where $\Lambda$ denotes the entire system. For a tripartite state, define the conditional mutual information by

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

where $S_{\rho_\beta}(X)$ is the von Neumann entropy of the reduced state on $X$. CMI decay at arbitrary temperatures. For general quantum Gibbs states $\rho_\beta$ at any temperature $\beta$, $I_{\rho_\beta}(A:C|B)$ decays rapidly (e.g., super-polynomially) with the distance between the arbitrary regions $A$ and $C$. This is a quantum analogue of the Hammersley–Clifford theorem for classical or commutative Hamiltonians. A conditional version follows under uniform clustering, while the general arbitrary-temperature statement remains an open question.