On Hidden States in Quantum Random Walks

TL;DR

This paper explores the existence and properties of hidden states in quantum random walks, linking them to hidden Markov processes and classical quantum mechanics debates, and establishing their ergodic and asymptotic behaviors.

Contribution

It unifies quantum random walks with hidden Markov and finitary processes, revealing their ergodic and asymptotic properties within a comprehensive model.

Findings

Quantum random walks are finitary processes with ergodic properties.

Hidden states in quantum walks relate to classical quantum mechanics debates.

Quantum walks exhibit quantum-style asymptotic behaviors.

Abstract

It was recently pointed out that identifiability of quantum random walks and hidden Markov processes underlie the same principles. This analogy immediately raises questions on the existence of hidden states also in quantum random walks and their relationship with earlier debates on hidden states in quantum mechanics. The overarching insight was that not only hidden Markov processes, but also quantum random walks are finitary processes. Since finitary processes enjoy nice asymptotic properties, this also encourages to further investigate the asymptotic properties of quantum random walks. Here, answers to all these questions are given. Quantum random walks, hidden Markov processes and finitary processes are put into a unifying model context. In this context, quantum random walks are seen to not only enjoy nice ergodic properties in general, but also intuitive quantum-style asymptotic properties. It is also pointed out how hidden states arising from our framework relate to hidden states in earlier, prominent treatments on topics such as the EPR paradoxon or Bell's inequalities.