Topological Characterization of Discrete-Time Classical Stochastic Processes: Dual Role of Point-Gap Topology
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909025721909248 |
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| author | Nakagawa, Masaya Ueda, Masahito |
| author_facet | Nakagawa, Masaya Ueda, Masahito |
| contents | We present topological characterization of classical stochastic processes described by discrete-time Markov chains on lattices. We point out that point-gap topology of stochastic matrices entails two distinct physical consequences that hinge on the choice of the reference point. The point-gap topology around a generic reference point is related to the direction of transport, and nontrivial topology around the origin of the complex spectrum of a stochastic matrix implies non-Markovianity caused by, e.g., feedback control. On the basis of this characterization, we identify the topological origin of directed transport in a classic experiment of Maxwell's demon [S. Toyabe et al., Nat. Phys. 6, 988 (2010)] and find the topological nature of feedback control beyond thermodynamic interpretation. We demonstrate that a topologically enforced non-Markovian classical stochastic process can be simulated by a Markovian quantum master equation, indicating a topological form of quantum advantage. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07231 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological Characterization of Discrete-Time Classical Stochastic Processes: Dual Role of Point-Gap Topology Nakagawa, Masaya Ueda, Masahito Mesoscale and Nanoscale Physics Statistical Mechanics Quantum Physics We present topological characterization of classical stochastic processes described by discrete-time Markov chains on lattices. We point out that point-gap topology of stochastic matrices entails two distinct physical consequences that hinge on the choice of the reference point. The point-gap topology around a generic reference point is related to the direction of transport, and nontrivial topology around the origin of the complex spectrum of a stochastic matrix implies non-Markovianity caused by, e.g., feedback control. On the basis of this characterization, we identify the topological origin of directed transport in a classic experiment of Maxwell's demon [S. Toyabe et al., Nat. Phys. 6, 988 (2010)] and find the topological nature of feedback control beyond thermodynamic interpretation. We demonstrate that a topologically enforced non-Markovian classical stochastic process can be simulated by a Markovian quantum master equation, indicating a topological form of quantum advantage. |
| title | Topological Characterization of Discrete-Time Classical Stochastic Processes: Dual Role of Point-Gap Topology |
| topic | Mesoscale and Nanoscale Physics Statistical Mechanics Quantum Physics |
| url | https://arxiv.org/abs/2605.07231 |