Realization of the probability laws in the quantum central limit theorems by a quantum walk

TL;DR

This paper explores the connection between quantum walks and quantum probability theory by deriving probability laws in quantum central limit theorems from a specific quantum walk model.

Contribution

It introduces a limit theorem for a non-localized initial state quantum walk and links it to quantum central limit theorems, revealing new insights into quantum probability.

Findings

Derived a limit theorem for a 2-state quantum walk with non-localized initial state

Generated probability laws in quantum central limit theorems from quantum walks

Established a potential role of quantum walks in quantum probability theory

Abstract

Since a limit distribution of a discrete-time quantum walk on the line was derived in 2002, a lot of limit theorems for quantum walks with a localized initial state have been reported. On the other hand, in quantum probability theory, there are four notions of independence (free, monotone, commuting, and boolean independence) and quantum central limit theorems associated to each independence have been investigated. The relation between quantum walks and quantum probability theory is still unknown. As random walks are fundamental models in the Kolmogorov probability theory, can the quantum walks play an important role in quantum probability theory? To discuss this problem, we focus on a discrete-time 2-state quantum walk with a non-localized initial state and present a limit theorem. By using our limit theorem, we generate probability laws in the quantum central limit theorems from the quantum walk.