Quantum Random Walks Based On Functionals Of Bernoulli Noises

Quantum Bernoulli noises are the family of annihilation and creation operators acting on Bernoulli functionals,which satisfy a canonical anti-commutation relation(CAR)in equal-time.In this paper,we first present some new results concerning quantum Bernoulli noises,which themselves are interesting.Then,based on these new results,we construct a time-dependent quantum random walk with infinitely many degrees of freedom.We prove that the walk has a unitary representation,hence belongs to the category of the so-called unitary quantum random walks.We examine its distribution property at the vacuum initial state and some other initial states and find that it has the same limit distribution as the classical random walk,which contrasts sharply with the case of the usual quantum random walks with finite degrees of freedom.This paper is organized as follows.In the first chapter,we describe the background and the state-of-art of the research concerning quantum random walks and our research topics.The second chapter gives fundamentals about Bernoulli noise functionals and quantum Bernoulli noises.Finally in the third chapter,we state and prove our main results.