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arXiv:2208.03770 · Open Systems & Information Dynamics

Open Quantum Random Walks and Quantum Markov chains on Trees I: Phase transitions

Farrukh Maksutovich Mukhamedov, Abdessatar Souissi, Tarek Hamdi

Published Mar 1, 2022• 15 citations
arXivHTMLPDF

Abstract

In the present paper, we construct QMC (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. However, we are going to look at the probability distribution as a Markov field over the Cayley tree. Such kind of consideration allows us to investigate phase transition phenomena associated for OQRW within QMC scheme. Furthermore, we first propose a new construction of QMC on trees, which is an extension of QMC considered in [10]. Using such a construction, we are able to construct QMCs on tress associated with OQRW. Our investigation leads to the detection of the phase transition phenomena within the proposed scheme. This kind of phenomena appears for the first time in this direction. Moreover, mean entropies of QMCs are calculated.

DOI
10.1142/s1230161222500032

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In the present paper, we construct QMC (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. However, we are going to look at the probability distribution as a Markov field over the Cayley tree. Such kind of consideration allows us to investigate phase transition phenomena associated for OQRW within QMC scheme. Furthermore, we first propose a new construction of QMC on trees, which is an extension of QMC considered in [10]. Using such a construction, we are able to construct QMCs on tress associated with OQRW. Our investigation leads to the detection of the phase transition phenomena within the proposed scheme. This kind of phenomena appears for the first time in this direction. Moreover, mean entropies of QMCs are calculated.
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.
Quantum Markov chains (QMCs) and open quantum random walks (OQRWs) represent different quantum extensions of the classical Markov chain framework. QMCs stand as a more profound layer within the realm of Markovian dynamics. The exploration of both QMCs and OQRWs has been a predominant focus over the past decade. Recently, a significant connection between QMCs and OQRWs has been forged, yielding valuable applications. This bridge is particularly impactful when studying QMCs on tree structures, where it intersects with the realm of phase transitions in models naturally arising from quantum statistical mechanics. Furthermore, it aids in elucidating statistical properties, such as recurrence and clustering. The objective of this paper centers around delving into the intricate structure of QMCs on Cayley trees in relation to OQRWs. The foundational elements of this class of QMCs are built upon using classical probability measures that encompass the hierarchical nature of Cayley trees. This exploration unveils the pivotal role that the dynamics of OQRWs play in shaping the behavior of the Markov chains under consideration. Moreover, the analysis extends to their classical counterparts. The findings are further underscored by the examination of notable examples, contributing to a comprehensive understanding of the outcomes.
In networks, the Markov clustering (MCL) algorithm is one of the most efficient approaches in detecting clustered structures. The MCL algorithm takes as input a stochastic matrix, which depends on the adjacency matrix of the graph network under consideration. Quantum clustering algorithms are proven to be superefficient over the classical ones. Motivated by the idea of a potential clustering algorithm based on quantum Markov chains, we prove a clustering property for quantum Markov chains (QMCs) on Cayley trees associated with open quantum random walks (OQRW).
This paper introduces a symmetric group dynamics on quantum Markov chains on Cayley trees (QMCCT). By applying symmetric group actions, we establish a novel algebraic perspective that enables the systematic study of local correlation functions in QMCCT. This approach provides a powerful tool for investigating phase transitions and characterizing ergodic properties in these systems. We further extend this framework to explore QMCCT associated with Open Quantum Random Walks (OQRWs). The integration of symmetric group dynamics allows for a detailed analysis of the interplay between group-theoretic symmetries and the stochastic processes underlying QMCCT, particularly in the context of OQRWs. Our results demonstrate the potential of this approach to uncover new insights into the structure and behaviour of quantum Markov chains.