Topological Characterization of Discrete-Time Classical Stochastic Processes: Dual Role of Point-Gap Topology

• URL: http://arxiv.org/abs/2605.07231v1
• Date: Fri, 08 May 2026 04:35:07 GMT
• Title: Topological Characterization of Discrete-Time Classical Stochastic Processes: Dual Role of Point-Gap Topology
• Abstract summary: We present topological characterization of classical processes described by discrete-time chains on lattices.<n>We identify the origin of directed transport in a classic experiment of Maxwell's demon.<n>We demonstrate that a topologically enforced non-Markovian classical process can be simulated by a Markovian quantum master equation.
• Score: 4.941630596191806
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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