Strong convergence of quantum random walks via semigroup decomposition

TL;DR

This paper presents a straightforward method to prove the strong convergence of quantum random walks to quantum stochastic cocycles, highlighting the algebraic structure and including applications to repeated quantum interactions.

Contribution

It introduces a simple, direct approach using semigroup decomposition to establish convergence and algebraic properties of quantum stochastic processes.

Findings

Quantum random walks converge strongly to quantum stochastic cocycles.

The approach reveals the algebraic structure of the convergence process.

Repeated quantum interactions are incorporated into the convergence scheme.

Abstract

We give a simple and direct treatment of the strong convergence of quantum random walks to quantum stochastic operator cocycles, via the semigroup decomposition of such cocycles. Our approach also delivers convergence of the pointwise product of quantum random walks to the quantum stochastic Trotter product of the respective limit cocycles, thereby revealing the algebraic structure of the limiting procedure. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.