Classical and quantum Markov networks—including Gibbs states of commuting local Hamiltonians—are characterized by the vanishing of conditional mutual information (CMI) between spatially separated subsystems. Adding local dissipation to a Markov network turns it into a hidden Markov network, in which CMI is not guaranteed to vanish even at long distances. Nevertheless, it is unclear whether the Markov property still holds approximately after dissipation, i.e., whether CMI decays with distance. In this work, we show that high-temperature hidden Markov networks are approximately Markovian under a wide class of local perturbations. We establish that CMI in high-temperature Gibbs states subject to local dissipation decays exponentially: (i) for classical Hamiltonians subject to arbitrary local transition matrices and (ii) for commuting local Hamiltonians subject to unital channels that obey certain mild restrictions. Our result provides the first rigorous understanding of long-range CMI in a broad class of models, offering a new lens to analyze quantum states through their information structure. It also leads to new results in understanding teleportation, mixed-state phases of matter, state preparation, and compressibility of quantum states.

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• Quantum information processing
• Quantum phase transitions
• Quantum teleportation
• Lindblad equation