Repeated Quantum Interactions and Unitary Random Walks

TL;DR

This paper characterizes discrete quantum evolution equations driven by classical randomness, showing they form random walks on the unitary group, with a focus on algebraic characterization of the underlying unitary operators.

Contribution

It provides an algebraic characterization of quantum systems driven by classical randomness, linking repeated interactions to unitary random walks.

Findings

Solutions are random walks on the unitary group

Characterization is algebraic in terms of the interaction's unitary operator

Identifies conditions for classical randomness in quantum evolutions

Abstract

Among the discrete evolution equations describing a quantum system $\rH_S$ undergoing repeated quantum interactions with a chain of exterior systems, we study and characterize those which are directed by classical random variables in $\RR^N$. The characterization we obtain is entirely algebraical in terms of the unitary operator driving the elementary interaction. We show that the solutions of these equations are then random walks on the group $U(\rH_0)$ of unitary operators on $\rH_0$.