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arXiv:2407.09384

Bi-Entangled Hidden Markov Processes and Recurrence

El Gheteb Soueidi

Published Jul 12, 2024• 0 citations
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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.

DOI
10.48550/arxiv.2407.09384

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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.
This paper consists of 333 parts. The first part only considers classical processes and introduces two different extensions of the notion of hidden Markov process. In the second part, the notion of quantum hidden process is introduced. In the third part it is proven that, by restricting various types of quantum Markov chains to appropriate commutative sub-algebras (diagonal sub-algebras) one recovers all the classical hidden process and, in addition, one obtains families of processes which are not usual hidden Markov process, but are included in the above mentioned extensions of these processes. In this paper we only deal with processes with an at most countable state space.
Starting from the notion of algebraic hidden processes introduced in Ref. 9 , we propose a definition of quantum hidden Markov process which extends the notion of quantum Markov chain (QMC) in the same sense in which classical hidden Markov processes (HMPs) extend classical Markov chains. We give several structure theorems for this new family of quantum states which shares with the QMCs the unique property that the finite dimensional joint expectations are explicitly given without these states being linearly equivalent to product states (as the Gaussian ones). The special feature of this new class of quantum states is best exemplified by looking at their restrictions to diagonal sub-algebras: we prove that they lead to a new family of classical stochastic processes that generalize in a non-trivial way the classical HMPs.
In this paper, we propose an algebraic approach to the definition of Hidden Markov Processes (HMP) which has two advantages: on the one hand, it prepares the ground for the quantum extension of these processes (discussed in the second part of this paper), on the other hand it suggests two natural extensions of these processes: one of them highlights a different notion of conditional independence between the observable and the hidden process; the other one drops the assumption that the hidden process is Markov. The latter extension is motivated by the examples of hidden, but not hidden Markov processes produced in the second part of the paper. Furthermore, we prove that any hidden Markov process can be interpreted as a restriction, to a suitable sub-algebra, of a Markov process.
We develop a rather general framework for constructing quantum Markov processes through Markov operator cocycles (see Chapter 2, Section 3) that satisfy a quantum stochastic differential equation.In order to achieve this goal we first recall the basic facts of Boson Fock quantum stochastic calculus and then give the fundamental results in the theory of quantum stochastic differential equations concerning existence, uniqueness, time reversal, isometricity and coisometricity of solutions. Next we construct the quantum flow associated with a Markov operator cocycle, and give a condition that guarantees that the restriction to a commutative subalgebra is a commutative flow.
In the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution [Formula: see text] of OQRW. This sheds new light on some properties of the measure [Formula: see text]. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of [Formula: see text]-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.