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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 $μ_n$ on the space of $n$-paths, and the $μ_n$ 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.