Abdessatar Barhoumi

• Full Professor
• Professor (Full) at King Faisal University

Quantum channels, pivotal in information processing, describe transformations within quantum systems and enable secure communication and error correction. Ergodic and mixing properties elucidate their behavior. In this paper, we establish a sufficient condition for mixing based on a quantum Markov-Dobrushin inequality. We prove that if the Markov-D...

We study a mixing condition for entangled Markov chains, so-called ψ-mixing property. We prove that every entangled Markov chain, whose Markov operators satisfy the Markov–Dobrushin condition, is ψ-mixing. Moreover, we show the restriction of the underlying QMC to the diagonal algebra gives rise to a classical mixing Markov chains with the same exp...

We clarify the structure of tree-homogeneous quantum Markov chains (THQMC) as a multi-dimensional quantum extension of homogeneous Markov chains. We provide a construction of a class of quantum Markov chains on the Cayley tree based on open quantum random walks. Moreover, we prove the uniqueness of THQMC for the construction under consideration, wh...

The problem of recurrence for quantum Markov chains on trees (QMCT), is more subtle than for 1D quantum Markov chains (QMC); it involves infinitely many rays due to the exponential growth of ramifications on the Cayley trees and their relevant constraints. We study criteria for recurrence of QMCT based on the correlations functions and boundary con...

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 analo...

The main aim of the present paper is to establish the existence of a phase transition for the quantum Ising model with competing XY interactions within the quantum Markov chain (QMC) scheme. In this scheme, we employ the $C^*$-algebraic approach to the phase transition problem. Note that these kinde of models do not have one-dimensional analogues,...

We express, in full generality, the Jacobi sequences and the orthogonal polynomials of the powers of a real-valued random variable (Formula presented.) with all moments, as functions of the corresponding sequences of the random variable itself.

In this paper we introduce a new notion of λ −order homogeneous operators on the nuclear algebra of white noise operators. Then, we give their Fock expansion in terms of quantum white noise (QWN) fields {at,at∗;t∈ℝ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...

The identification mentioned in the title allows a formulation of the multidi mensional Favard Lemma different from the ones currently used in the literature and which exactly parallels the original one dimensional formulation in the sense that the positive Jacobi sequence is replaced by a sequence of positive Hermitean (square) matrices and the re...

In this paper, we consider the Ising-XY model with competing interactions on the Cayley tree of order two. This model can be seen as a non-commutative (i.e. J-XY -interactions on next-neighbor vertices) perturbation of the classical Ising model on the Cayley tree. For the considered model we establish the existence of three translation-invariant qu...

In the paper Accardi et al.: Identification of the theory of orthogonal polynomials in d–indeterminates with the theory of 3–diagonal symmetric interacting Fock spaces on \(\mathbb {C} ^d\), submitted to: IDA–QP (Infinite Dimensional Anal. Quantum Probab. Related Topics), [1], it has been shown that, with the natural definitions of morphisms and is...

We express the Jacobi sequences of the square of a real valued random variable with all moments, not necessarily symmetric, as functions of the corresponding sequences of the random variable itself. In the symmetric case, the result is known and, we give a short, purely algebraic proof of it. We apply our result to the square of the Gamma distribut...

It is known that the disordered phase of the classical Ising model on the Caley tree is extreme in some region of the temperature. If one considers the Ising model with competing interactions on the same tree, then about the extremity of the disordered phase there is no any information. In the present paper, we first aiming to analyze the correspon...

We prove that, each probability meassure on ℝ, with all moments, is canonically associated with (i) a∗-Lie algebra; (ii) a complexity index labeled by pairs of natural integers. The measures with complexity index (0,K) consist of two disjoint classes: that of all measures with finite support and the semi-circle-arcsine class (the discussion in Sec....

The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC...

The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC...

We introduce a new notion of quantum white noise (QWN) first-order Wick differential operators with variable coefficients on the topological nuclear algebra ℒ(ℱθ(N'), ℱθ*(N')) endowed with the Wick product and we give their chaos expansions in terms of the QWN-derivatives {Dz-, Dz+ : z ∈ N}. We also study a generalized transport equation associated...

In this paper we develop a theory of quantum Laplacians

we develop an operator theory on a nuclear algebra of white noise operators in terms of the quantum white noise (QWN) derivatives and their dual adjoints. Using an adequate definition of a QWN-symbol transformation, we discuss QWN-integral-sum kernel operators which give the Fock expansion of the QWN-operators (i.e. the linear operators acting on n...

In this paper we introduce a quantum white noise (QWN) convolution calculus over a nuclear algebra of operators. We use this calculus to discuss new solutions of some linear and non-linear differential equations.

The main objective of this paper is to investigate an extension of the "Volterra-Gross" Laplacian on nuclear algebra of generalized functions. In so doing, without using the renormalization procedure, this extension provides a continuous nuclear realization of the square white noise Lie algebra obtained by Accardi–Franz–Skeide in Ref. 2. An extende...

The quantum white noise (QWN)-Euler operator Δ E Q is defined as the sum Δ G Q +N Q , where Δ G Q and N Q stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that Δ E Q has an integral representation in terms of the QWN-derivatives {D t - ,D t + ;t∈ℝ} as a kind of functional integra...

In this paper, a theory of stochastic processes generated by quantum extensions of Laplacians is developed. Representations of the associated heat semigroups are discussed by means of suitable time shifts. In particular the quantum Brownian motion associated to the Lévy–Laplacian is obtained as the usual Volterra–Gross Laplacian using the Cesàro Hi...

Consider the Lévy–Meixner one-mode interacting Fock space {ΓLM, 〈 ⋅, ⋅ 〉LM}. Inspired by a derivative formula appearing in 〈 ⋅, ⋅ 〉LM, we define scalar products 〈 ⋅, ⋅ 〉LM, n in symmetric n-particle spaces. Then, we introduce a class of one-mode type interacting Fock spaces naturally associated to the one-dimensional infinitely divisible distributi...

By adapting the white noise theory, the quantum analogues of the (classical) Gross Laplacian and Lévy Laplacian, the so-called quantum Gross Laplacian and quantum Lévy Laplacian, respectively, are introduced as the Laplacians acting on the spaces of generalized operators. Then the integral representations of the quantum Laplacians are studied in te...

We introduce a one-mode type interacting Fock space {F}NB ( {H}) naturally associated to the negative binomial distribution μr,α. The Fourier transform in generalized joint eigenvectors of a family {Jϕ ; ϕ ∈ ɛ} of Pascal Jacobi fields provides a way to explicit a unitary isomorphism {U}{r,α } between {F}NB ( {H}) and the so-called Pascal white nois...

By using an appropriate one-mode type interacting Fock spaces, Γ M (ℋ), introduced in [L. Accardi, A. Barhoumi, and A. Riahi, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13, No. 3, 435–460 (2010; Zbl 1225.60121)], we define a nuclear triple ℱ M,θ (ℰ)⊂L 2 (ℰ ' ,Λ M )⊂ℱ M,θ * (ℰ) of test and generalized functions, with θ being a suitable Young f...

In the present paper, we extend the notion of quantum time shift, and the related results obtained in [5], from representations of current algebras of the Heisenberg Lie algebra to representations of current algebras of the Oscillator Lie algebra. This produces quantum extensions of a class of classical Lévy processes much wider than the usual Brow...

The main purpose of this paper is to investigate a generalized oscillator algebra, naturally associated to the Lévy-Meixner polynomials and a class of nonlinear coherent vector. We derive their overcompleteness relation, in so doing, the partition of the unity in terms of the eigenstates of the sequences of coherent vectors is established. An examp...

Based on nuclear algebra of entire functions, we extend some results about operator-parameter transforms involving the Fourier-Gauss and Fourier-Mehler transforms. We investigate the solution of a initial-value problem associated to infinitesimal generators of these transformations. In particular, by using convolution product, we show to what exten...

In this paper white noise analysis with respect to the Lévy process with negative binomial distributed marginals is investigated. An appropriate space of distributions, ′, is used to describe the structure of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure ΛNB. The constructed decomposition is us...

By using an appropriate space of distributions, , we derive the chaos decomposition property of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure ΛNB. The constructed decomposition is used to define a nuclear triple of test and generalized functions, where θ is a Young function satisfying some suit...

A new concept of q-symmetric tensor product is defined, where q is a parameter in the interval (-1,1]. Duality theorems are established for new spaces called generalized q-Fock spaces and some of their features are indicated.

We introduce a new Hilbert space HL, called Levy{Hilbert space, arising from the Levy Laplacian. This Hilbert space is used to construct a new Gel'fand triple called Levy{Gel'fand triple, which exhibits the essentially inflnite dimensional character of the Levy Laplacian. By using the fact that the usual trace and the Levy trace (being associated w...

We study a quantum extension of the Lévy Laplacian, so-called quantum Lévy-type Laplacian, to the nuclear algebra of operators on spaces of entire functions. We give several examples of the action of the quantum Lévy-type Laplacian on basic operators and we study a quantum white noise convolution differential equation involving the quantum Lévy-typ...

In this paper we introduce a new scalar product on distribution spaces based on the Cesàro mean of a sequence. We then use this scalar product to construct a family of separable Hilbert spaces H C , called Cesàro Hilbert spaces and naturally associated to the Lévy Laplacian. Finally we use the essentially infinite dimensional character of the Lévy...

In this paper we study the Gross heat equation perturbed by noises with the initial condition being a generalized function. The noises are given by either a white noise or a space-time white noise. The main technique we use is the representation of the Gross Laplacian as a convolution operator. It enables us to apply the convolution calculus on a s...

The main purpose of this paper is to investigate a generalized oscillator algebra, naturally associated to the Gegenbauer polynomials and a related class of nonlinear coherent vectors. We derive their overcomplete-ness relation, in so doing, the partition of unity in terms of the eigenstates of the sequences of coherent vectors is established. It t...