Bi-Entangled Hidden Markov Processes and Recurrence

TL;DR

This paper introduces Bi-entangled hidden Markov processes, exploring their quantum entanglement properties, recurrence behavior, and how restricting to certain subalgebras recovers classical Markov processes.

Contribution

It defines Bi-entangled hidden Markov processes, provides a joint expectation formula, and analyzes their recurrence and classical reduction within quantum frameworks.

Findings

Joint expectation formula for Bi-entangled processes

Recurrence properties of quantum Markov processes

Reduction to classical Markov processes via commutative subalgebras

Abstract

In this paper, we introduce the notion of Bi-entangled hidden Markov processes. These are hidden quantum processes where the hidden processes themselves exhibit entangled Markov process, and the observable processes also exhibit entanglement. We present a specific formula for the joint expectation of these processes. Furthermore, we discuss the recurrence of the underlying quantum Markov processes associated to the Bi-entangled hidden Markov processes and we establish that, by restricting them within suitable commutative subalgebras (diagonal subalgebras) leads to the recovery of Markov processes defined by the hidden stochastic matrix. In this paper we only deal with processes with an at most countable state space.