The Random Walk in Generalized Quantum Theory

TL;DR

This paper generalizes the classical random walk within a framework of generalized quantum theory, introducing a decoherence functional that allows for pairwise interference and maintains Markovian and translational invariance properties.

Contribution

It develops a new formulation of quantum random walks by defining a decoherence functional under strong positivity, extending classical probability to quantum interference scenarios.

Findings

Established a class of Markovian, translationally invariant decoherence functionals

Demonstrated the conditions for strong positivity in generalized quantum walks

Extended classical random walk models to include quantum interference effects

Abstract

One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a certain condition of ``strong positivity'', the most general Markovian, translationally invariant ``decoherence functional'' with nearest neighbor transitions.