Probabilistic stability of magnetic quantum walk

A magnetic quantum walk is a quantum walk coupling to (or perturbed by) a magnetic field, which plays an active role in describing the motion of a particle in a magnetic field. In this paper, we introduce and investigate a discrete-time magnetic quantum walk in the framework of Bernoulli functionals. We first introduce magnetic shift operators in terms of the annihilation and creation operators on Bernoulli functionals, and then use them to construct the desired walk. After proving some technical results, we establish a formula for determining probability distributions of the walk. Finally, we show that the walk has probability distributions that are completely independent of the magnetic potential when its initial state meets some mild conditions.