Abdessatar Souissi - Academia.edu (original) (raw)

Papers by Abdessatar Souissi

Research paper thumbnail of Nonlinear Stochastic Operators and Associated Inhomogeneous Entangled Quantum Markov Chains

Journal of nonlinear mathematical physics, Feb 26, 2024

In the present paper, we introduce a class of F-stochastic operators on a finitedimensional simpl... more In the present paper, we introduce a class of F-stochastic operators on a finitedimensional simplex, each of which is regular, ascertaining that the species distribution in the succeeding generation corresponds to the species distribution in the previous one in the long run. It is proposed a new scheme to define nonhomogeneous Markov chains contingent on the F-stochastic operators and given initial data. By means of the uniform ergodicity of the non-homogeneous Markov chain, we define a non-homogeneous (quantum) entangled Markov chain. Furthermore, it is established that the non-homogeneous entangled Markov chain enables-mixing property.

Research paper thumbnail of Clustering quantum Markov chains on trees associated with open quantum random walks

AIMS Mathematics

In networks, the Markov clustering (MCL) algorithm is one of the most efficient approaches in det... more 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).

Research paper thumbnail of Entangled Hidden Markov Models

Chaos, Solitons & Fractals

Research paper thumbnail of Hidden processes and hidden Markov processes: classical and quantum

arXiv (Cornell University), Feb 14, 2023

Research paper thumbnail of On a ψ-Mixing property for Entangled Markov Chains

Physica A: Statistical Mechanics and its Applications

Research paper thumbnail of Clustering property for quantum Markov chains on the comb graph

AIMS Mathematics

Quantum Markov chains (QMCs) on graphs and trees were investigated in connection with many import... more 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 −Isingmodeloverthecombgraph-Ising model over the comb graph −Isingmodeloverthecombgraph \mathbb N\rhd_0 \mathbb Z $. Mainly, we prove that the QMC associated with the disordered phase, enjoys a clustering property.

Research paper thumbnail of On stopping rules for tree-indexed quantum Markov chains

Infinite Dimensional Analysis, Quantum Probability and Related Topics

In the present paper, we introduce stopping rules and related notions for quantum Markov chains o... more In the present paper, we introduce stopping rules and related notions for quantum Markov chains on trees (QMCT). We prove criteria for recurrence, accessibility and irreducibility for QMCT. This work extends to trees the notion of stopping times for quantum Markov chains (QMC) introduced by Accardi and Koroliuk, which plays a key role in the study of many properties of QMC. Moreover, we illustrate the obtained results for a concrete model of XY-Ising type.

Research paper thumbnail of Tree-Homogeneous Quantum Markov Chains

International Journal of Theoretical Physics

Research paper thumbnail of Open quantum random walks and quantum Markov Chains on trees II: the recurrence

Quantum Information Processing, Jun 3, 2023

In the present paper, we construct QMC (Quantum Markov Chains) associated with Open Quantum Rando... more 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 of OQRW. Furthermore, we first propose a new construction of QMC on trees, which is an extension of QMC considered in Ref. [9]. 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 first time in this direction. Moreover, mean entropies of QMCs are calculated.

Research paper thumbnail of Review of: "Why Backward Time Travel Is Not Possible

The idea of traveling back in time is quite interesting. Your analysis is based on the experiment... more The idea of traveling back in time is quite interesting. Your analysis is based on the experimental impossibility of localizing a particle's position in space-time if it has irrational location. In this case, the provided argumentation is quite convincing. However, from a mathematical point of view, it is possible to find an infinite path of four-vectors with integer magnitudes. From Lagrange's four-square theorem, every natural number can be represented as the sum of four squares of integers. In particular, every square number is the sum of four squares. This invites us to think about a possible backward time travel machile.

Research paper thumbnail of Recurrence of a class of quantum Markov chains on trees

Chaos Solitons & Fractals, Nov 1, 2022

Research paper thumbnail of Block Markov Chains on Trees

Cornell University - arXiv, Aug 23, 2020

We introduce block Markov chains (BMCs) indexed by an infinite rooted tree. It turns out that BMC... more We introduce block Markov chains (BMCs) indexed by an infinite rooted tree. It turns out that BMCs define a new class of tree-indexed Markovian processes. We clarify the structure of BMCs in connection with Markov chains (MCs) and Markov random fields (MRFs). Mainly, show that probability measures which are BMCs for every root are indeed Markov chains (MCs) and yet they form a strict subclass of Markov random fields (MRFs) on the considered tree. Conversely, a class of MCs which are BMCs is characterized. Furthermore, we establish that in the one-dimensional case the class of BMCs coincides with MCs. However, a slight perturbation of the one-dimensional lattice leads to us to an example of BMCs which are not MCs appear.

Research paper thumbnail of Entropy of quantum Markov states on Cayley trees

Journal of Statistical Mechanics: Theory and Experiment

In this paper, we continue the investigation of quantum Markov states (QMSs) and define their mea... more In this paper, we continue the investigation of quantum Markov states (QMSs) and define their mean entropies. Such entropies are explicitly computed under certain conditions. The present work takes a huge leap forward at tackling one of the most important open problems in quantum probability, which concerns the calculations of mean entropies of quantum Markov fields. Moreover, it opens up a new perspective for the generalization of many interesting results related to the one-dimensional QMSs and quantum Markov chains to multi-dimensional cases.

Research paper thumbnail of A Markov–Dobrushin Inequality for Quantum Channels

Open Systems & Information Dynamics

We propose a quantum extension of the Markov-Dobrushin inequality. As an application, we estimate... more We propose a quantum extension of the Markov-Dobrushin inequality. As an application, we estimate the Markov-Dobrushin constant for some classes of quantum Markov channels, in particular for the Pauli channel, widely studied in quantum information theory.

Research paper thumbnail of On the Limit of Superposition States

arXiv: Mathematical Physics, Oct 31, 2020

In this paper, we study the structure of a family of superposition states on tensor algebras. The... more In this paper, we study the structure of a family of superposition states on tensor algebras. The correlation functions of the considered states are described through a new kind of positive definite kernels valued in the dual of C *-algebras, so-called Schur kernels. Mainly, we show the existence of the limiting state of a net of superposition states over an arbitrary locally finite graph. Furthermore, we show that this limiting state enjoys a mixing property and an α-mixing property in the case of the multi-dimensional integer lattice Z ν .

Research paper thumbnail of A Forward Quantum Markov Field on Graphs

arXiv: Mathematical Physics, 2020

In this paper, we propose a class of quantum Markov fields QMF on a graphs G=(V,E)G= (V,E)G=(V,E). The Markov... more In this paper, we propose a class of quantum Markov fields QMF on a graphs G=(V,E)G= (V,E)G=(V,E). The Markov structure of the considered QMF is investigated in the finer structure of a quasi-local algebrav mathcalAV\mathcal{A}_VmathcalAV of observables based over a graphs GGG. Namely, the considered Markovian fields are infinite volume states defined through a generating couple (varphi(0),(mathcalEycupNy))(\varphi^{(0)}, (\mathcal{E}_{\{y\}\cup N_y}))(varphi(0),(mathcalEycupNy)) of a product state varphi(0)\varphi^{(0)}varphi(0) on mathcalAV\mathcal{A}_VmathcalAV and a family of local transition expectations mathcalEycupNy\mathcal{E}_{\{y\}\cup N_y}mathcalEycupNy based on a vertex yyy and the set of it nearest-neighbors. The main result of the paper concerns the existence and the uniqueness of QMF associated with a couple (varphi(0),(mathcalEycupNy))(\varphi^{(0)}, (\mathcal{E}_{\{y\}\cup N_y}))(varphi(0),(mathcalEycupNy)) for on an important class of graphs including trees strictly.

Research paper thumbnail of A Class of Quantum Markov Fields on Tree-like Graphs: Ising-type Model on a Husimi Tree

Open Systems & Information Dynamics, 2021

A new class of forward quantum Markov fields (FQMFs) is introduced. The structure of these quantu... more A new class of forward quantum Markov fields (FQMFs) is introduced. The structure of these quantum Markov fields is investigated in the finer structure of a quasi-local algebra of observable over a tree-like graph. We provide an effective construction of a class of FQMCs. Moreover, we show the existence of three FMRFs associated with an Ising type model on a Husimi tree.

Research paper thumbnail of Open Quantum Random Walks and Quantum Markov Chains

Functional Analysis and Its Applications, 2019

In the present paper we construct quantum Markov chains associated with open quantum random walks... more In the present paper we construct quantum Markov chains associated with open quantum random walks in the sense that the transition operator of a chain is determined by an open quantum random walk and the restriction of the chain to the commutative subalgebra coincides with the distribution P ρ of the walk. This sheds new light on some properties of the measure P ρ. For example, this measure can be considered as the distribution of some functions of a certain Markov process.

Research paper thumbnail of Supercyclic and Hypercyclic Generalized Weighted Backward Shifts over a Non-Archimedean c0(N) Space

Mathematics, Nov 22, 2021

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Research paper thumbnail of Quantum Markov Chains on the Comb graphs: Ising model

In the present paper, we construct quantum Markov chains (QMC) over the Comb graphs. As an applic... more In the present paper, we construct quantum Markov chains (QMC) over the Comb graphs. As an application of this construction, it is proved the existence of the disordered phase for the Ising type models (within QMC scheme) over the Comb graphs. Moreover, it is also established that the associated QMC has clustering property with respect to translations of the graph. We stress that this paper is the ?rst one where a nontrivial example of QMC over non-regular graphs is given.

Research paper thumbnail of Nonlinear Stochastic Operators and Associated Inhomogeneous Entangled Quantum Markov Chains

Journal of nonlinear mathematical physics, Feb 26, 2024

In the present paper, we introduce a class of F-stochastic operators on a finitedimensional simpl... more In the present paper, we introduce a class of F-stochastic operators on a finitedimensional simplex, each of which is regular, ascertaining that the species distribution in the succeeding generation corresponds to the species distribution in the previous one in the long run. It is proposed a new scheme to define nonhomogeneous Markov chains contingent on the F-stochastic operators and given initial data. By means of the uniform ergodicity of the non-homogeneous Markov chain, we define a non-homogeneous (quantum) entangled Markov chain. Furthermore, it is established that the non-homogeneous entangled Markov chain enables-mixing property.

Research paper thumbnail of Clustering quantum Markov chains on trees associated with open quantum random walks

AIMS Mathematics

In networks, the Markov clustering (MCL) algorithm is one of the most efficient approaches in det... more 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).

Research paper thumbnail of Entangled Hidden Markov Models

Chaos, Solitons & Fractals

Research paper thumbnail of Hidden processes and hidden Markov processes: classical and quantum

arXiv (Cornell University), Feb 14, 2023

Research paper thumbnail of On a ψ-Mixing property for Entangled Markov Chains

Physica A: Statistical Mechanics and its Applications

Research paper thumbnail of Clustering property for quantum Markov chains on the comb graph

AIMS Mathematics

Quantum Markov chains (QMCs) on graphs and trees were investigated in connection with many import... more 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 −Isingmodeloverthecombgraph-Ising model over the comb graph −Isingmodeloverthecombgraph \mathbb N\rhd_0 \mathbb Z $. Mainly, we prove that the QMC associated with the disordered phase, enjoys a clustering property.

Research paper thumbnail of On stopping rules for tree-indexed quantum Markov chains

Infinite Dimensional Analysis, Quantum Probability and Related Topics

In the present paper, we introduce stopping rules and related notions for quantum Markov chains o... more In the present paper, we introduce stopping rules and related notions for quantum Markov chains on trees (QMCT). We prove criteria for recurrence, accessibility and irreducibility for QMCT. This work extends to trees the notion of stopping times for quantum Markov chains (QMC) introduced by Accardi and Koroliuk, which plays a key role in the study of many properties of QMC. Moreover, we illustrate the obtained results for a concrete model of XY-Ising type.

Research paper thumbnail of Tree-Homogeneous Quantum Markov Chains

International Journal of Theoretical Physics

Research paper thumbnail of Open quantum random walks and quantum Markov Chains on trees II: the recurrence

Quantum Information Processing, Jun 3, 2023

In the present paper, we construct QMC (Quantum Markov Chains) associated with Open Quantum Rando... more 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 of OQRW. Furthermore, we first propose a new construction of QMC on trees, which is an extension of QMC considered in Ref. [9]. 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 first time in this direction. Moreover, mean entropies of QMCs are calculated.

Research paper thumbnail of Review of: "Why Backward Time Travel Is Not Possible

The idea of traveling back in time is quite interesting. Your analysis is based on the experiment... more The idea of traveling back in time is quite interesting. Your analysis is based on the experimental impossibility of localizing a particle's position in space-time if it has irrational location. In this case, the provided argumentation is quite convincing. However, from a mathematical point of view, it is possible to find an infinite path of four-vectors with integer magnitudes. From Lagrange's four-square theorem, every natural number can be represented as the sum of four squares of integers. In particular, every square number is the sum of four squares. This invites us to think about a possible backward time travel machile.

Research paper thumbnail of Recurrence of a class of quantum Markov chains on trees

Chaos Solitons & Fractals, Nov 1, 2022

Research paper thumbnail of Block Markov Chains on Trees

Cornell University - arXiv, Aug 23, 2020

We introduce block Markov chains (BMCs) indexed by an infinite rooted tree. It turns out that BMC... more We introduce block Markov chains (BMCs) indexed by an infinite rooted tree. It turns out that BMCs define a new class of tree-indexed Markovian processes. We clarify the structure of BMCs in connection with Markov chains (MCs) and Markov random fields (MRFs). Mainly, show that probability measures which are BMCs for every root are indeed Markov chains (MCs) and yet they form a strict subclass of Markov random fields (MRFs) on the considered tree. Conversely, a class of MCs which are BMCs is characterized. Furthermore, we establish that in the one-dimensional case the class of BMCs coincides with MCs. However, a slight perturbation of the one-dimensional lattice leads to us to an example of BMCs which are not MCs appear.

Research paper thumbnail of Entropy of quantum Markov states on Cayley trees

Journal of Statistical Mechanics: Theory and Experiment

In this paper, we continue the investigation of quantum Markov states (QMSs) and define their mea... more In this paper, we continue the investigation of quantum Markov states (QMSs) and define their mean entropies. Such entropies are explicitly computed under certain conditions. The present work takes a huge leap forward at tackling one of the most important open problems in quantum probability, which concerns the calculations of mean entropies of quantum Markov fields. Moreover, it opens up a new perspective for the generalization of many interesting results related to the one-dimensional QMSs and quantum Markov chains to multi-dimensional cases.

Research paper thumbnail of A Markov–Dobrushin Inequality for Quantum Channels

Open Systems & Information Dynamics

We propose a quantum extension of the Markov-Dobrushin inequality. As an application, we estimate... more We propose a quantum extension of the Markov-Dobrushin inequality. As an application, we estimate the Markov-Dobrushin constant for some classes of quantum Markov channels, in particular for the Pauli channel, widely studied in quantum information theory.

Research paper thumbnail of On the Limit of Superposition States

arXiv: Mathematical Physics, Oct 31, 2020

In this paper, we study the structure of a family of superposition states on tensor algebras. The... more In this paper, we study the structure of a family of superposition states on tensor algebras. The correlation functions of the considered states are described through a new kind of positive definite kernels valued in the dual of C *-algebras, so-called Schur kernels. Mainly, we show the existence of the limiting state of a net of superposition states over an arbitrary locally finite graph. Furthermore, we show that this limiting state enjoys a mixing property and an α-mixing property in the case of the multi-dimensional integer lattice Z ν .

Research paper thumbnail of A Forward Quantum Markov Field on Graphs

arXiv: Mathematical Physics, 2020

In this paper, we propose a class of quantum Markov fields QMF on a graphs G=(V,E)G= (V,E)G=(V,E). The Markov... more In this paper, we propose a class of quantum Markov fields QMF on a graphs G=(V,E)G= (V,E)G=(V,E). The Markov structure of the considered QMF is investigated in the finer structure of a quasi-local algebrav mathcalAV\mathcal{A}_VmathcalAV of observables based over a graphs GGG. Namely, the considered Markovian fields are infinite volume states defined through a generating couple (varphi(0),(mathcalEycupNy))(\varphi^{(0)}, (\mathcal{E}_{\{y\}\cup N_y}))(varphi(0),(mathcalEycupNy)) of a product state varphi(0)\varphi^{(0)}varphi(0) on mathcalAV\mathcal{A}_VmathcalAV and a family of local transition expectations mathcalEycupNy\mathcal{E}_{\{y\}\cup N_y}mathcalEycupNy based on a vertex yyy and the set of it nearest-neighbors. The main result of the paper concerns the existence and the uniqueness of QMF associated with a couple (varphi(0),(mathcalEycupNy))(\varphi^{(0)}, (\mathcal{E}_{\{y\}\cup N_y}))(varphi(0),(mathcalEycupNy)) for on an important class of graphs including trees strictly.

Research paper thumbnail of A Class of Quantum Markov Fields on Tree-like Graphs: Ising-type Model on a Husimi Tree

Open Systems & Information Dynamics, 2021

A new class of forward quantum Markov fields (FQMFs) is introduced. The structure of these quantu... more A new class of forward quantum Markov fields (FQMFs) is introduced. The structure of these quantum Markov fields is investigated in the finer structure of a quasi-local algebra of observable over a tree-like graph. We provide an effective construction of a class of FQMCs. Moreover, we show the existence of three FMRFs associated with an Ising type model on a Husimi tree.

Research paper thumbnail of Open Quantum Random Walks and Quantum Markov Chains

Functional Analysis and Its Applications, 2019

In the present paper we construct quantum Markov chains associated with open quantum random walks... more In the present paper we construct quantum Markov chains associated with open quantum random walks in the sense that the transition operator of a chain is determined by an open quantum random walk and the restriction of the chain to the commutative subalgebra coincides with the distribution P ρ of the walk. This sheds new light on some properties of the measure P ρ. For example, this measure can be considered as the distribution of some functions of a certain Markov process.

Research paper thumbnail of Supercyclic and Hypercyclic Generalized Weighted Backward Shifts over a Non-Archimedean c0(N) Space

Mathematics, Nov 22, 2021

This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY

Research paper thumbnail of Quantum Markov Chains on the Comb graphs: Ising model

In the present paper, we construct quantum Markov chains (QMC) over the Comb graphs. As an applic... more In the present paper, we construct quantum Markov chains (QMC) over the Comb graphs. As an application of this construction, it is proved the existence of the disordered phase for the Ising type models (within QMC scheme) over the Comb graphs. Moreover, it is also established that the associated QMC has clustering property with respect to translations of the graph. We stress that this paper is the ?rst one where a nontrivial example of QMC over non-regular graphs is given.