Abstract:

Quantum Bernoulli noises (QBN) are annihilation and creation operators acting on the space of square integrable Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal time and can play an important role in describing the environment of an open quantum system. In this paper, we address a type of perturbations of the canonical unitary involutions associated with QBN. We analyze these perturbations from a perspective of spectral theory and obtain exactly their spectra, which coincide with their point spectra. We also discuss eigenvectors of these perturbations from an algebraic point of view and unveil the structures of the subspaces consisting of their eigenvectors. Finally, as application, we consider the abstract quantum walks driven by these perturbations and obtain infinitely many invariant probability distributions of these walks.

Key words: Quantum probability, Quantum Bernoulli noise, Perturbation of unitary involution, Spectrum, Quantum walk