Two-Site Quantum Random Walk

TL;DR

This paper investigates the measure-theoretic properties of a two-site quantum random walk, focusing on extending the quantum measure to relevant subsets of path space and characterizing the quantum integral.

Contribution

It introduces a method to extend the quantum measure to a quadratic algebra of path subsets and provides a new characterization of the quantum integral for the model.

Findings

Quantum measure can be extended to a quadratic algebra of subsets.

The quantum measure cannot be extended to the full sigma-algebra.

A new characterization of the quantum integral is provided.

Abstract

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure $\mu_n$ on the space of $n$-paths, and the $\mu_n$ in turn induce a quantum measure $\mu$ on the cylinder sets within the space $\Omega$ of untruncated paths. Although $\mu$ cannot be extended to a continuous quantum measure on the full $\sigma$-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of $\Omega$ in a systematic way. We begin an investigation of this problem by showing that $\mu$ can be extended to a quantum measure on a "quadratic algebra" of subsets of $\Omega$ that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the $n$-path space.