Random walks in finite Abelian groups with Birkhoff subpolytopes of doubly stochastic matrices and their physical implementation

TL;DR

This paper explores random walks on finite Abelian groups using Markov chains with doubly stochastic matrices, analyzing their properties and physical implementations in quantum systems, including the additive and Heisenberg-Weyl groups.

Contribution

It introduces a novel framework linking random walks in finite Abelian groups with Birkhoff polytopes and provides physical implementations using quantum measurement techniques.

Findings

Probability vectors form a shrinking polytope over time.

Quantitative measures like Lorenz values and Gini index describe the distributions.

Physical implementations demonstrated for specific groups.

Abstract

Random walks in a finite Abelian group $G$ are studied. They use Markov chains with doubly stochastic transition matrices, in a Birkhoff subpolytope ${\cal B}(G)$ associated with the group $G$. It is shown that all future probability vectors belong to a polytope which does not depend on the transition matrices, and which shrinks during time evolution. Various quantities are used to describe the probability vectors: the majorization preorder, Lorenz values and the Gini index, entropic quantities, and the total variation distance. The general results are applied to the additive group ${\mathbb Z}(d)$, and to the Heisenberg-Weyl group $HW(d)/{\mathbb Z}(d)$. A physical implementation of random walks in ${\mathbb Z}(d)$ that involves a sequence of non-selective projective measurements, is discussed. A physical implementation of random walks in the Heisenberg-Weyl group $HW(d)/{\mathbb Z}(d)$ using a sequence of non-selective POVM measurements with coherent states, is also presented.