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AIMS Mathematics

Clustering property for quantum Markov chains on the comb graph

Abdessatar Souissi, El Gheteb Soueidi, Mohamed Rhaima

Published Jan 1, 2023• 6 citations
Publisher

Abstract

Quantum Markov chains (QMCs) on graphs and trees were investigated in connection with many important models arising from quantum statistical mechanics and quantum information. These quantum states generate many important properties such as quantum phase transition and clustering properties. In the present paper, we propose a construction of QMCs associated with an $ XX $-Ising model over the comb graph $ \mathbb N\rhd_0 \mathbb Z $. Mainly, we prove that the QMC associated with the disordered phase, enjoys a clustering property.

DOI
10.3934/math.2023396

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Quantum Markov chains (QMCs) on graphs and trees were investigated in connection with many important models arising from quantum statistical mechanics and quantum information. These quantum states generate many important properties such as quantum phase transition and clustering properties. In the present paper, we propose a construction of QMCs associated with an XXXXXX-Ising model over the comb graph N⊳0Z\mathbb N\rhd_0 \mathbb ZN⊳0​Z. Mainly, we prove that the QMC associated with the disordered phase, enjoys a clustering property.
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).
The main aim of the present paper by means of the quantum Markov chain (QMC) approach is to establish the existence of a phase transition for the quantum Ising model with competing XY interaction. In this scheme, the C*-algebraic approach is employed to the phase transition problem. Note that these kinds of models do not have one-dimensional analogs, i.e., the considered model persists only on trees. It turns out that if the Ising part interactions vanish, then the model with only competing XY-interactions on the Cayley tree of order two does not have a phase transition. By phase transition, we mean the existence of two distinct QMCs that are not quasi-equivalent and their supports do not overlap. Moreover, it is also shown that the QMC associated with the model has a clustering property, which implies that the von Neumann algebras corresponding to the states are factors.
The main aim of the present paper is to provide a new construction of quantum Markov chain (QMC) on arbitrary order Cayley tree. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Note that this construction reminds statistical mechanics models with competing interactions on trees. If one considers one dimensional tree, then the provided construction reduces to well-known one, which was studied by the first author. Our construction will allow to investigate phase transition problem in a quantum setting.
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 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.