Bounding the convergence time of local probabilistic evolution

TL;DR

This paper develops a general bound on the convergence time of local probabilistic evolutions using isoperimetric inequalities, applicable to classical and quantum systems, enhancing understanding of their convergence behavior.

Contribution

It introduces a novel approach to bounding convergence times of local probabilistic evolutions via isoperimetric inequalities, extending to non-Markovian and quantum dynamics.

Findings

Derived a new convergence bound using isoperimetric inequalities.

Extended bounds to quantum dynamics beyond Markovian models.

Provided insights into the connectedness and convergence properties of state spaces.

Abstract

Isoperimetric inequalities form a very intuitive yet powerful characterization of the connectedness of a state space, that has proven successful in obtaining convergence bounds. Since the seventies they form an essential tool in differential geometry, graph theory and Markov chain analysis. In this paper we use isoperimetric inequalities to construct a bound on the convergence time of any local probabilistic evolution that leaves its limit distribution invariant. We illustrate how this general result leads to new bounds on convergence times beyond the explicit Markovian setting, among others on quantum dynamics.