A Random Walk in Physics

Jean‐Christophe Pain

The purpose of this article is to highlight the connections between two seemingly distinct domains: random walks and the distribution of angular-momentum projections in quantum physics (the magnetic quantum numbers m). It is well known that there is indeed a deep mathematical link between the two, via the vector composition of angular momenta and rotational symmetry. Random walks are considered in the framework of an interpretation of the probability of microstates in statistical physics. The ideas presented in this work aim to illustrate the relevance of this perspective for modeling angular momentum in atomic physics.

Cécile DeWitt-Morette

Viv M. Kendon, Barry C. Sanders

We show that quantum walks interpolate between a coherent ``wave walk'' and a random walk depending on how strongly the walker's coin state is measured; i.e., the quantum walk exhibits the quintessentially quantum property of complementarity, which is manifested as a tradeoff between knowledge of which path the walker takes vs the sharpness of the interference pattern. A physical implementation of a quantum walk (the quantum quincunx) should thus have an identifiable walker and the capacity to demonstrate the interpolation between wave walk and random walk depending on the strength of measurement.

Dietrick E. Thomsen

Apostol Vourdas

Abstract Random walks in a finite Abelian group G are studied. They use Markov chains with doubly stochastic transition matrices, in a Birkhoff subpolytope 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 Z ( d ) , and to the Heisenberg–Weyl group H W ( d ) / Z ( d ) . A physical implementation of random walks in Z ( d ) that involves a sequence of non-selective projective measurements, is discussed. A physical implementation of random walks in the Heisenberg–Weyl group H W ( d ) / Z ( d ) using a sequence of non-selective positive operator-valued measure measurements with coherent states, is also presented.

Giuliano G. La Guardia