Decoherence for Markov chains

TL;DR

This paper explores the algebraic structure of the peripheral spectrum subspace of stochastic matrices, framing the restriction as a $C^*$-dynamical system to understand decoherence in Markov chains.

Contribution

It introduces a novel $C^*$-algebraic framework for analyzing the peripheral spectrum of stochastic matrices and their associated dynamical systems.

Findings

The peripheral spectrum subspace forms a finite-dimensional abelian $C^*$-algebra.

Restriction of the stochastic matrix to this subspace generates a conservative $C^*$-dynamical system.

Provides a new algebraic perspective on decoherence phenomena in Markov chains.

Abstract

The subspace generated by the eigenvectors pertaining to the peripheral spectrum of any stochastic matrix is naturally equipped with a structure of a (finite dimensional abelian) $C^*$-algebra, and the restriction of such a stochastic matrix to this subspace, indeed a $C^*$-algebra under this canonical new product, generates a conservative $C^*$-dynamical system.