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Infinite Dimensional Analysis Quantum Probability and Related Topics

Hidden quantum Markov processes

Luigi Accardi, El Gheteb Soueidi, Yun Gang Lu, Abdessatar Souissi

Published May 9, 2024• 9 citations
Publisher

Abstract

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.

DOI
10.1142/s0219025724500073

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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.
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.
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.
This paper extended the framework of quantum Markovianity by introducing backward and inverse backward quantum Markov chains (QMCs). We established the existence of these models under general conditions, demonstrating their applicability to a wide range of quantum systems. Our findings revealed distinct structural properties within these models, providing new insights into their dynamics and relationships to finitely correlated states. These advancements contributed to a deeper understanding of quantum processes and have potential implications for various quantum applications, including hidden quantum Markov processes.
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.
In this work, we extend the idea of quantum Markov chains (Gudder in J Math Phys 49(7):072105 [ 3 ]) in order to propose quantum hidden Markov models (QHMMs). For that, we use the notions of transition operation matrices and vector states, which are an extension of classical stochastic matrices and probability distributions. Our main result is the Mealy QHMM formulation and proofs of algorithms needed for application of this model: Forward for general case and Vitterbi for a restricted class of QHMMs. We show the relations of the proposed model to other quantum HMM propositions and present an example of application.