Two-site Quantum Random Walk

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