Improved estimate of local Markovianity for quantum Gibbs states

Abstract: We establish improved locally Markovian properties of quantum Gibbs states at every fixed positive temperature. For finite-range interactions of bounded interaction degree, the conditional mutual information (CMI) decays exponentially with separation, with a prefactor exponential in the boundary size. This improves the exponential volume dependence of the local Markov bounds established by Chen and Rouzé \cite{Chen2025}. For exponentially and stretched-exponentially decaying interactions, we obtain stretched-exponential CMI decay with a boundary-dependent prefactor. For power-law interactions, we prove inverse-logarithmic or algebraic decay with polynomial region-size dependence. Our proof is static, in contrast to the dynamical approach of Chen and Rouzé. We compare the Gibbs state with a product state obtained by removing boundary interactions, reducing the CMI bound to a relative-entropy loss that we control using locality estimates.