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arXiv:2606.28054

Static features from mixing in short- and long-range Lindbladians: Markov property and correlations

Paul Rosa-Ruiz, Matteo Scandi, Ángela Capel, Álvaro M. Alhambra

quant-phcond-mat.stat-mechmath-phSubmitted Jun 26, 2026• Announced Jun 29, 2026• 0 citations
v1 · Jun 26, 2026v2 · Jul 7, 2026
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Abstract

The classification of mixed-state phases requires criteria beyond two-point correlation functions, such as the decay of the mutual information (MI) and the conditional mutual information (CMI), with the latter encapsulated in the notion of Markov length. Here we show how such static properties of the fixed point of a Lindbladian follow from natural dynamical features of its generators: rapid mixing and frustration-freeness. We focus on systems with long-range interactions, and prove (i) that local Lindbladians satisfying (global) rapid mixing and frustration-freeness have fixed-points whose CMI decays with the shielding distance, and (ii) that (local) rapid mixing together with primitivity and regularity implies global decay of MI. For long-range interactions decaying with a power law with rate $α$, both quantities decay polynomially rather than exponentially, in contrast to the finite- and short-range regimes where exponential decay (a finite Markov length) is expected within a phase. We further show that Gibbs states of long-range, non-commuting Hamiltonians satisfy a local Markov property at any temperatures, extending the recent results (Chen--Rouzé, 2025) for short-range systems to the long-range regime relevant to a variety of experimental platforms. As a numerical example, we study the long-range Ising model both with and without a transverse field. We find regimes in which the polynomial decay of the CMI holds, in accordance with the bounds proven.

Comments
54 pages, 4 figures, comments welcome

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The classification of mixed-state phases requires criteria beyond two-point correlation functions, such as the decay of the mutual information (MI) and the conditional mutual information (CMI), with the latter encapsulated in the notion of Markov length. Here we show how such static properties of the fixed point of a Lindbladian follow from natural dynamical features of its generators: rapid mixing and frustration-freeness. We focus on systems with long-range interactions, and prove (i) that local Lindbladians satisfying (global) rapid mixing and frustration-freeness have fixed-points whose CMI decays with the shielding distance, and (ii) that (local) rapid mixing together with primitivity and regularity implies global decay of MI. For long-range interactions decaying with a power law with rate ααα, both quantities decay polynomially rather than exponentially, in contrast to the finite- and short-range regimes where exponential decay (a finite Markov length) is expected within a phase. We further show that Gibbs states of long-range, non-commuting Hamiltonians satisfy a local Markov property at any temperatures, extending the recent results (Chen--Rouzé, 2025) for short-range systems to the long-range regime relevant to a variety of experimental platforms. As a numerical example, we study the long-range Ising model both with and without a transverse field. We find regimes in which the polynomial decay of the CMI holds, in accordance with the bounds proven.
For quantum phases of Hamiltonian ground states, the energy gap plays a central role in ensuring the stability of the phase as long as the gap remains finite. We propose Markov length, the length scale at which the quantum conditional mutual information (CMI) decays exponentially, as an equally essential quantity characterizing mixed-state phases and transitions. For a state evolving under a local Lindbladian, we argue that if its Markov length remains finite along the evolution, then it remains in the same phase, meaning there exists another quasilocal Lindbladian evolution that can reverse the former one. We apply this diagnostic to toric code subject to decoherence and show that the Markov length is finite everywhere except at its decodability transition, at which it diverges. CMI in this case can be mapped to the free energy cost of point defects in the random bond Ising model. This implies that the mixed-state phase transition coincides with the decodability transition and also suggests a quasilocal decoding channel.
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