Two-Site Quantum Random Walk

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.

28 pages

References in corpus (6)

• Quantum Dynamics without the Wave Function
• An exercise in "anhomomorphic logic"
• Quantum measure and integration theory
• On extending the Quantum Measure
• Hilbert Spaces from Path Integrals
• An anhomomorphic logic for quantum mechanics

Cited by in corpus (4)

• Quantum walks: a comprehensive review
• How to Measure the Quantum Measure
• Does a Quantum Particle Know its Own Energy?
• A "problem of time" in the multiplicative scheme for the $n$-site hopper