Quantum random walks and minors of Hermitian Brownian motion

TL;DR

This paper constructs discrete-time quantum random walk approximations for eigenvalue processes of minors of Hermitian Brownian motion, extending Markov property results to certain noncommutative submatrices.

Contribution

It establishes the Markov property for eigenvalues of specific submatrices of Hermitian Brownian motion in the noncommutative setting, generalizing previous results.

Findings

Discrete-time approximations of eigenvalue processes constructed

Markov property proven for certain submatrix eigenvalues

Extension of Markov property results to noncommutative case

Abstract

Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam and van Moerbeke that the process of eigenvalues of two consecutive minors of an Hermitian Brownian motion is a Markov process, whereas if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.