Geometrical, topological, and dynamical description of N interacting spin- s particles in a long-range Ising model and their interplay with quantum entanglement

Brahim Amghar, Abdallah Slaoui, Jamal Elfakir +1 more

arXiv:2210.16578Physical Review AMar 2, 202311 citations

EPJ Web of Conferences

A Conformal Geometric Approach to Quantum Entanglement for Spin-1/2 Particles

F. De Martini, Enrico Santamato

Published Jan 1, 2013• 1 citation

Publisher

Abstract

DOI: 10.1051/epjconf/20135801012

Related papers

Unveiling the "mystery" of quantum nonlocality by conformal geometrodynamics

F. De Martini, Enrico Santamato

The problem of quantum entanglement of two spin-1/2 particles is faced in a conformally invariant geometric framework. The configuration space of the two particles is extended by adding orientational degrees of freedom and quantum effects, including entanglement, are derived from the conformal curvature of this space. A mechanism is proposed where the space curvature and the particle motion are in mutual interaction and it is proved that this feedback between geometry and dynamics reproduces all quantum features of the two-particle system. Entanglement, in particular, originates from the residual nonlocal interaction among the orientational degrees of freedom of the two spinning particles.The problem of quantum entanglement of two spin-1/2 particles is faced in a conformally invariant geometric framework. The configuration space of the two particles is extended by adding orientational degrees of freedom and quantum effects, including entanglement, are derived from the conformal curvature of this space. A mechanism is proposed where the space curvature and the particle motion are in mutual interaction and it is proved that this feedback between geometry and dynamics reproduces all quantum features of the two-particle system. Entanglement, in particular, originates from the residual nonlocal interaction among the orientational degrees of freedom of the two spinning particles.

AIP conference proceedingsJan 1, 20122 citations

SOLVING THE NONLOCALITY RIDDLE BY CONFORMAL QUANTUM GEOMETRODYNAMICS

Enrico Santamato, F. De Martini

Since the 1935 proposal by Einstein, Podolsky and Rosen the riddle of nonlocality, today demonstrated by the violation of Bell's inequalities within innumerable experiments, has been a cause of concern and confusion within the debate over the foundations of quantum mechanics. The present paper tackles the problem by a nonrelativistic approach based on conformal differential geometry applied to the solution of the dynamical problem of two entangled spin 1/2 particles. It is found that the quantum nonlocality may be understood on the basis of a conformal quantum geometrodynamics acting necessarily on the full "configuration space" of the entangled particles. At the end, the violation of the Bell inequalities is demonstrated without making recourse to the common nonlocality paradigm.Since the 1935 proposal by Einstein, Podolsky and Rosen the riddle of nonlocality, today demonstrated by the violation of Bell's inequalities within innumerable experiments, has been a cause of concern and confusion within the debate over the foundations of quantum mechanics. The present paper tackles the problem by a nonrelativistic approach based on conformal differential geometry applied to the solution of the dynamical problem of two entangled spin 1/2 particles. It is found that the quantum nonlocality may be understood on the basis of a conformal quantum geometrodynamics acting necessarily on the full "configuration space" of the entangled particles. At the end, the violation of the Bell inequalities is demonstrated without making recourse to the common nonlocality paradigm.

International Journal of Quantum InformationDec 1, 20124 citations

Solving the nonlocality riddle by conformal quantum geometrodynamics

Enrico Santamato, F. De Martini

A rigorous ab initio derivation of the quantum mechanics of a single particle with spin is presented starting by conformally Weyl-gauge-invariant principle. The particle is described as a relativistic top with six Euler's angles and quantum effects are introduced by assuming that the Weyl's curvature of the particle configuration space acts on it as an external scalar potential. Weyl's conformal covariance is made explicit in all steps of the theory. It is shown that metric of the configuration space accounts for non-quantum relativistic effects, while the affine connections account for quantum relativistic effects. In this way, classical and quantum features acquire well distinguished geometrical origin. A scalar wave function is also introduced to recover the connection with the standard quantum description based on Dirac's four-component spinors. Finally, the case of two entangled spins is considered in the nonrelativistic limit and it is found that the nonlocality rests on the entanglement of the spin internal orientational variables, playing the role of "hidden variables". The theory was carried out in the Minkowski space-time, but it can be easily extended to a space with nonzero Riemann curvature.A rigorous ab initio derivation of the quantum mechanics of a single particle with spin is presented starting by conformally Weyl-gauge-invariant principle. The particle is described as a relativistic top with six Euler's angles and quantum effects are introduced by assuming that the Weyl's curvature of the particle configuration space acts on it as an external scalar potential. Weyl's conformal covariance is made explicit in all steps of the theory. It is shown that metric of the configuration space accounts for non-quantum relativistic effects, while the affine connections account for quantum relativistic effects. In this way, classical and quantum features acquire well distinguished geometrical origin. A scalar wave function is also introduced to recover the connection with the standard quantum description based on Dirac's four-component spinors. Finally, the case of two entangled spins is considered in the nonrelativistic limit and it is found that the nonlocality rests on the entanglement of the spin internal orientational variables, playing the role of "hidden variables". The theory was carried out in the Minkowski space-time, but it can be easily extended to a space with nonzero Riemann curvature.

Journal of Physics Conference SeriesJun 10, 20131 citation

Quantum Entanglement Geometry

Jincheng Zhang

Quantum entanglement, a phenomenon where two or more particles become linked regardless of distance, has captivated physicists and philosophers alike. This research explores the potential of utilizing geometric principles to understand and manipulate entanglement, proposing a novel model to represent quantum entanglement as a geometric structure. We aim to investigate how the inherent geometry of entangled states can be leveraged to optimize quantum computation and communication protocols. The core claim is that establishing a geometric model of quantum entanglement will unlock new avenues for enhancing the performance of these technologies. This paper delves into the foundational concepts and proposes a mechanism for translating entanglement into a geometric framework, ultimately seeking to reveal hidden properties and potential optimizations within the space of entangled quantum states.Quantum entanglement, a phenomenon where two or more particles become linked regardless of distance, has captivated physicists and philosophers alike. This research explores the potential of utilizing geometric principles to understand and manipulate entanglement, proposing a novel model to represent quantum entanglement as a geometric structure. We aim to investigate how the inherent geometry of entangled states can be leveraged to optimize quantum computation and communication protocols. The core claim is that establishing a geometric model of quantum entanglement will unlock new avenues for enhancing the performance of these technologies. This paper delves into the foundational concepts and proposes a mechanism for translating entanglement into a geometric framework, ultimately seeking to reveal hidden properties and potential optimizations within the space of entangled quantum states.

Aug 31, 20260 citations

Quantum Entanglement Curvature Field

Jincheng Zhang

This paper introduces the Quantum Entanglement Curvature Field (QECF) model, proposing that entangled quantum spaces exhibit curvature analogous to spacetime curvature. The core concept revolves around the mathematical representation of this curvature, denoted as [R_E], where R_E represents the entanglement curvature. This model seeks to establish a geometric framework for understanding entanglement, potentially leading to a new field of study – entanglement geometry physics. The exploration focuses on multi-particle quantum systems and aims to provide a foundational understanding for future research in this area. The model's potential implications are significant, suggesting a fundamental shift in our understanding of quantum mechanics and its relationship to geometry. ---This paper introduces the Quantum Entanglement Curvature Field (QECF) model, proposing that entangled quantum spaces exhibit curvature analogous to spacetime curvature. The core concept revolves around the mathematical representation of this curvature, denoted as [R_E], where R_E represents the entanglement curvature. This model seeks to establish a geometric framework for understanding entanglement, potentially leading to a new field of study – entanglement geometry physics. The exploration focuses on multi-particle quantum systems and aims to provide a foundational understanding for future research in this area. The model's potential implications are significant, suggesting a fundamental shift in our understanding of quantum mechanics and its relationship to geometry. ---

Aug 17, 20260 citations

Geometrical, topological, and dynamical description of N interacting spin- s particles in a long-range Ising model and their interplay with quantum entanglement

Brahim Amghar, Abdallah Slaoui, Jamal Elfakir +1 more

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 the 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 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.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 the 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 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.

arXiv:2210.16578Physical Review AMar 2, 202311 citations

\mathcal{N}

interacting spin-

\mathtt{s}

under the 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 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.