Homogeneous Open Quantum Random Walks on a Lattice

We study open quantum random walks (OQRWs) for which the underlying graph is a lattice, and the generators of the walk are homogeneous in space. Using the results recently obtained in Carbone and Pautrat (Ann Henri Poincaré, 2015), we study the quantum trajectory associated with the OQRW, which is described by a position process and a state process. We obtain a central limit theorem and a large deviation principle for the position process. We study in detail the case of homogeneous OQRWs on the lattice $$\mathbb {Z}^d$$Zd, with internal space $$\mathfrak {h}=\mathbb {C}^2$$h=C2.