arXiv subjects

Jamal Elfakir

Publications and source records attributed to Jamal Elfakir.

3 recordsLinked to original sources

arXiv2026

Exploring the Geometric and Dynamical Properties of Spin Systems and Their Interplay with Quantum Entanglement

This thesis, explores the quantum entanglement and evolution through both a geometric and dynamical perspective. The first part focuses on classical phase space and its central role in Hamiltonian mechanics, emphasizing the importance of symplectic structures in describing mechanical states. The study highlights the formal analogy between classical phase space and the Hilbert space used in quantum mechanics. The second part is devoted to the geometric description of quantum states through the projective structure of Hilbert space. Emphasis is placed on the geometric interpretation of quantum evolution, particularly via the Fubini-Study metric, associated symplectic structures, and the geometric phase acquired during unitary evolutions. The final two parts are dedicated to the study of spin systems (both two-body and many-body) under different interaction models (XXZ Heisenberg and all-range Ising). Both the dynamical aspects (evolution speed, entanglement, and the quantum brachistochrone problem) and the geometric and topological structures of the corresponding quantum states are analyzed.

quant-ph↗

arXiv2024

Complementarity between quantum entanglement, geometrical and dynamical appearances in N spin-$1/2$ system under all-range Ising model

With the growth of geometric science, including the methods of exploring the world of information by means of modern geometry, there has always been a mysterious and fascinating ambiguous link between geometric, topological and dynamical characteristics with quantum entanglement. Since geometry studies the interrelations between elements such as distance and curvature, it provides the information sciences with powerful structures that yield practically useful and understandable descriptions of integrable quantum systems. We explore here these structures in a physical system of $N$ interaction spin-$1/2$ under all-range Ising model. By performing the system dynamics, we determine the Fubini-Study metric defining the relevant quantum state space. Applying Gaussian curvature within the scope of the Gauss-Bonnet theorem, we proved that the dynamics happens on a closed two-dimensional manifold having both a dumbbell-shape structure and a spherical topology. The geometric and topological phases appearing during the system evolution processes are sufficiently discussed. Subsequently, we resolve the quantum brachistochrone problem by achieving the time-optimal evolution. By restricting the whole system to a two spin-$1/2$ system, we investigate the relevant entanglement from two viewpoints; The first is of geometric nature and explores how the entanglement level affects derived geometric structures such as the Fubini-Study metric, the Gaussian curvature, and the geometric phase. The second is of dynamic nature and addresses the entanglement effect on the evolution speed and the related Fubini-Study distance. Further, depending on the degree of entanglement, we resolve the quantum brachistochrone problem.

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arXiv2023

Geometrical, topological and dynamical description of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model and their interplay with quantum entanglement

Comprehending the connections between the geometric, topological, and dynamical structures of integrable quantum systems with quantum phenomena exploitable in quantum information tasks, such as quantum entanglement, is a major problem in geometric information science. In this work we investigate these issues in a physical system of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model. We discover the relevant dynamics, identify the corresponding quantum phase space and we derive the associated Fubini-Study metric. Through the application of the Gauss-Bonnet theorem and the derivation of the Gaussian curvature, we have proved that the dynamics occurs on a spherical topology manifold. Afterwards, we analyze the gained geometrical phase under the arbitrary and cyclic evolution processes and solve the quantum brachistochrone problem by establishing the time-optimal evolution. Moreover, by narrowing the system to a two spin-$\mathtt{s}$ system, we explore the relevant entanglement from two different perspectives; The first is geometrical in nature and involves the investigation of the interplay between the entanglement degree and the geometrical structures, such as the Fubini-Study metric, the Gaussian curvature and the geometrical phase. The second is dynamical in nature and tackles the entanglement effect on the evolution speed and geodesic distance. Additionally, we resolve the quantum brachistochrone problem based on the entanglement degree.

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