On a generalized Central Limit Theorem and Large Deviations for Homogeneous Open Quantum Walks

TL;DR

This paper extends the central limit theorem and large deviations principles to homogeneous open quantum walks, providing explicit formulas and removing previous restrictions on the walk's properties.

Contribution

It generalizes existing CLT results for open quantum walks, offering explicit asymptotic expressions without additional conditions, and establishes large deviations principles.

Findings

Position process asymptotically approaches a mixture of Gaussian measures

Provides explicit formulas for asymptotic quantities

Establishes large deviations principles for certain cases

Abstract

We consider homogeneous open quantum random walks on a lattice with finite dimensional local Hilbert space and we study in particular the position process of the quantum trajectories of the walk. We prove that the properly rescaled position process asymptotically approaches a mixture of Gaussian measures. We can generalize the existing central limit type results and give more explicit expressions for the involved asymptotic quantities, dropping any additional condition on the walk. We use deformation and spectral techniques, together with reducibility properties of the local channel associated with the open quantum walk. Further, we can provide a large deviations' principle in the case of a fast recurrent local channel and at least lower and upper bounds in the general case.