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We prove that the quantum Gibbs states of spin systems above a certainthreshold temperature are approximate quantum Markov networks, meaning that theconditional mutual information decays rapidly with distance. We demonstrate theexponential decay for short-ranged interacting systems and power-law decay forlong-ranged interacting systems. Consequently, we establish the efficiency ofquantum Gibbs sampling algorithms, a strong version of the area law, thequasi-locality of effective Hamiltonians on subsystems, a clustering theoremfor mutual information, and a polynomial-time algorithm for classical Gibbsstate simulations.

Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature