Higher-Dimensional Open Quantum Walk Constructed from Quantum Bernoulli Noises

Abstract

Quantum Bernoulli noises are annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relations (CAR) in equal-time. In this paper, we use quantum Bernoulli noises to introduce a model of open quantum walk on the $d$-dimensional integer lattice $\mathbb{Z}^d$ for a general positive integer $d\geq 2$, which we call the $d$-dimensional open QBN walk. We obtain a quantum channel representation of the $d$-dimensional open QBN walk, and find that it admits the ``separability-preserving'' property. We prove that, for a wide range of choices of its initial state, the $d$-dimensional open QBN walk has a limit probability distribution of $d$-dimensional Gauss type. Finally we unveil links between the $d$-dimensional open QBN walk and the unitary quantum walk recently introduced in [Ce Wang and Caishi Wang, Higher-dimensional quantum walk in terms of quantum Bernoulli noises, Entropy 2020, 22, 504].