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arXiv:2609.38007

Improved estimate of local Markovianity for quantum Gibbs states

Tai Hsuan Yang

quant-phmath-phSubmitted Sep 29, 2026• Announced Sep 30, 2026• 0 citations• Revised Oct 5, 2026 (v2)
v1 · Sep 29, 2026v2 · Oct 5, 2026
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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 of only one of the two regions and independent of the size of the other region and the total system. 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.

Comments
35 pages, no figures, improved exposition

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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 of only one of the two regions and independent of the size of the other region and the total system. 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.
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Conditional mutual information (CMI) probes spatial non-Markovianity and plays a central role in a recently developed framework for defining mixed-state phases based on local reversibility. This framework sharply distinguishes exponentially decaying from algebraically decaying CMI, since the latter obstructs equivalence to a state with finite Markov length. Here we show that states that flow to the same renormalization-group (RG) fixed point can nevertheless exhibit qualitatively different CMI. Starting from a critical Gibbs state of a finite-range local classical Hamiltonian, whose CMI vanishes beyond the interaction range, we show that an RG-irrelevant perturbation that breaks detailed balance generically generates algebraically decaying CMI. Crucially, these power-law tails are cutoff-suppressed: although they lead to an infinite Markov length at every fixed lattice cutoff, they vanish as a positive power of the cutoff in the continuum limit. We construct solvable models that exhibit this phenomenon and predict that the Ising-symmetric critical point of Toom's cellular automata generically exhibits such behavior. We also find analogous cutoff-suppressed CMI in the high-temperature paramagnetic phase of the long-range Ising paramagnet, in contrast with the genuine power-law CMI at the finite-temperature critical point in this same model that survives the continuum limit. We also derive a general result that in one spatial dimension, CMI decays faster than the inverse square of the buffer length if and only if the long-distance physics is Markovian.
Conditional mutual information (CMI) is a widely studied multipartite correlation measure with a connection to the Markov property. Intriguingly, CMI can be generated by local dissipation, a phenomenon closely related to teleportation. The onset of long-range CMI corresponds to an information-theoretic mixed-state phase transition, with far-ranging implications for phase transitions, decoding, and state compressions. Little is known, however, about the conditions under which dissipation can generate long-range CMI. In this Letter, we provide the first rigorous results in this direction. We establish that CMI in high-temperature commuting Gibbs states subject to local dissipation decays exponentially under certain mild restrictions. Conversely, we show that the same setup can sustain long-range CMI at low temperatures. Our results establish the existence of finite-temperature information-theoretic phase transitions even in models that have no finite-temperature thermodynamic phase transitions.