Quantum Markov conjecture for thermal states

Let $\rho_\beta$ be an arbitrary quantum Gibbs state on a system partitioned as $\Lambda=A\sqcup E\sqcup B$. Let ${\mathcal{I}}_{\rho_\beta}(A:B|E)$ denote the conditional mutual information of $A$ and $B$ conditioned on $E$, and let $d_{A,B}$ be the distance between $A$ and $B$.

Quantum Markov conjecture. For arbitrary quantum Gibbs states,

$${\mathcal{I}}_{\rho_\beta}(A:B|E)\leq {\rm poly}(|A|,|B|)e^{-d_{A,B}/\xi_\beta},$$

with $\xi_\beta={\rm poly}(\beta)$.

This approximate quantum Markov property would give exponential clustering of conditional mutual information at arbitrary temperatures. The source reports that the conjecture had previously been proved only in high-temperature regimes, so its general status is recorded here as solved according to the supplied resolution evidence.