  We prove that the quantum Gibbs states of spin systems above a certain
threshold temperature are approximate quantum Markov networks, meaning that the
conditional mutual information decays rapidly with distance. We demonstrate the
exponential decay for short-ranged interacting systems and power-law decay for
long-ranged interacting systems. Consequently, we establish the efficiency of
quantum Gibbs sampling algorithms, a strong version of the area law, the
quasi-locality of effective Hamiltonians on subsystems, a clustering theorem
for mutual information, and a polynomial-time algorithm for classical Gibbs
state simulations.
