Quantum Approximate Markov Chains are Thermal

We prove an upper bound on the conditional mutual information of Gibbs states of one-dimensional short-range quantum Hamiltonians at finite temperature. We show the mutual information between two regions A and C conditioned on a middle region B decays exponentially with the square root of the length of B. Conversely, we also prove that any one-dimensional quantum state with small conditional mutual information in all tripartite splits of the line can be well-approximated by a Gibbs state of a local quantum Hamiltonian. These two results constitute a variant for one-dimensional quantum systems of the Hammersley-Clifford theorem (which characterizes Markov networks, i.e. probability distributions which have vanishing conditional mutual information, as Gibbs states of classical Hamiltonians). The result can be seen as a strengthening -- for one-dimensional systems -- of the mutual information area law for thermal states. It directly implies a method to efficiently prepare any one-dimensional Gibbs state at finite temperature by a constant-depth quantum circuit.