A Dobrushin condition for quantum Markov chains: Rapid mixing and conditional mutual information at high temperature
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914084393320448 |
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| author | Bakshi, Ainesh Liu, Allen Moitra, Ankur Tang, Ewin |
| author_facet | Bakshi, Ainesh Liu, Allen Moitra, Ankur Tang, Ewin |
| contents | A central challenge in quantum physics is to understand the structural properties of many-body systems, both in equilibrium and out of equilibrium. For classical systems, we have a unified perspective which connects structural properties of systems at thermal equilibrium to the Markov chain dynamics that mix to them. We lack such a perspective for quantum systems: there is no framework to translate the quantitative convergence of the Markovian evolution into strong structural consequences.
We develop a general framework that brings the breadth and flexibility of the classical theory to quantum Gibbs states at high temperature. At its core is a natural quantum analog of a Dobrushin condition; whenever this condition holds, a concise path-coupling argument proves rapid mixing for the corresponding Markovian evolution. The same machinery bridges dynamic and structural properties: rapid mixing yields exponential decay of conditional mutual information (CMI) without restrictions on the size of the probed subsystems, resolving a central question in the theory of open quantum systems. Our key technical insight is an optimal transport viewpoint which couples quantum dynamics to a linear differential equation, enabling precise control over how local deviations from equilibrium propagate to distant sites. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08542 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Dobrushin condition for quantum Markov chains: Rapid mixing and conditional mutual information at high temperature Bakshi, Ainesh Liu, Allen Moitra, Ankur Tang, Ewin Quantum Physics Data Structures and Algorithms A central challenge in quantum physics is to understand the structural properties of many-body systems, both in equilibrium and out of equilibrium. For classical systems, we have a unified perspective which connects structural properties of systems at thermal equilibrium to the Markov chain dynamics that mix to them. We lack such a perspective for quantum systems: there is no framework to translate the quantitative convergence of the Markovian evolution into strong structural consequences. We develop a general framework that brings the breadth and flexibility of the classical theory to quantum Gibbs states at high temperature. At its core is a natural quantum analog of a Dobrushin condition; whenever this condition holds, a concise path-coupling argument proves rapid mixing for the corresponding Markovian evolution. The same machinery bridges dynamic and structural properties: rapid mixing yields exponential decay of conditional mutual information (CMI) without restrictions on the size of the probed subsystems, resolving a central question in the theory of open quantum systems. Our key technical insight is an optimal transport viewpoint which couples quantum dynamics to a linear differential equation, enabling precise control over how local deviations from equilibrium propagate to distant sites. |
| title | A Dobrushin condition for quantum Markov chains: Rapid mixing and conditional mutual information at high temperature |
| topic | Quantum Physics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.08542 |