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

TL;DR

This paper uses point-gap topology to analyze classical stochastic processes, revealing how topological features relate to transport direction, non-Markovianity, and quantum advantages.

Contribution

It introduces a topological framework for classical stochastic processes, linking topology to physical phenomena like transport and feedback control, and explores quantum simulation implications.

Findings

Point-gap topology relates to transport direction in stochastic matrices.

Nontrivial topology around zero spectrum indicates non-Markovianity.

Topologically enforced non-Markovian processes can be simulated quantum mechanically.

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.