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

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.