Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1507.02197 · Journal of Physics A Mathematical and Theoretical

Geometry of a two-spin quantum state in evolution

Andrij Romanovych Kuzmak, V M Tkachuk

Published Dec 18, 2015• 15 citations
arXivHTMLPDF

Abstract

Abstract We study the quantum evolution of a two-spin system described by the isotropic Heisenberg Hamiltonian in the external magnetic field. It is shown that this evolution happens on a two-parametric closed manifold. The Fubini–Study metric of this manifold is obtained. It is found that this is the metric of the torus. The entanglement of the states which belong to this manifold is investigated.

DOI
10.1088/1751-8113/49/4/045301

Related papers

Geometrical and dynamical description of two interacting spins under the XXZ-type Heisenberg model

Jamal Elfakir, Brahim Amghar, Mohammed Daoud A

We consider a physical system consisting of two interacting spins governed by the [Formula: see text]-type Heisenberg Hamiltonian in an external magnetic field. We investigate the quantum evolution and the Riemannian geometry of the two-spin state space by means of the relevant Fubini–Study metric. The geometrical phase accumulated by the two-spin state is also examined under arbitrary and cyclic evolutions. By computing the evolution speed and the corresponding geodesic distance, we solve the quantum brachistochrone problem. The entanglement between the two spins is also studied via two approaches: the first one deals with the entanglement effect on the Fubini–Study metric and the geometrical phase, while the second one treats the entanglement effect on the evolution speed and the corresponding geodesic distance. Finally, we solve the quantum brachistochrone problem using the entanglement degree.

International Journal of Geometric Methods in Modern Physics•Aug 26, 2022•4 citations

Quantum state manifold and geometric, dynamic and topological phases for an interacting two-spin system

Brahim Amghar, Mohammed Daoud A

We consider a two-spin system of [Formula: see text] Heisenberg type submitted to an external magnetic field. Using the associated [Formula: see text] geometry, we investigate the dynamics of the system. We explicitly give the corresponding Fubini–Study metric. We show that for arbitrary pure initial states, the dynamics occurs on a torus. We compute the geometric phase, the dynamic phase and the topological phase. We investigate the interplay between the torus geometry and the entanglement of the two spins. In this respect, we provide a detailed analysis of the geometric phase, the dynamics velocity and the geodesic distance measured by the Fubini–Study metric in terms of the degree of entanglement between the two spins.

International Journal of Geometric Methods in Modern Physics•Dec 16, 2019•8 citations

Probing the geometry of two-qubit state space by evolution

Andrzej M. Frydryszak, Maria Gieysztor, Andrij Romanovych Kuzmak

We provide explicit geometric description of state manifolds obtained from evolution governed by a four-parameter family of time-independent Hamiltonians. We cover most cases related to the real interacting two-qubit systems and discuss possible types of evolutions in terms of the defining parameters. The relevant description of the pure state spaces and their Riemannian geometry with the Fubini–Study metric is given. In particular, we analyze the modification of known geometry of quantum state manifold by the linear noncommuting perturbation of the Hamiltonian. Finally, we investigate the behavior of the entanglement for obtained families of states resulting from the unitary evolution.

arXiv:1809.00731•Quantum Information Processing•Feb 6, 2019•6 citations

Geometry and quantum brachistochrone analysis of multiple entangled spin-1/2 particles under all-range Ising interaction

B. Amghar, M. Yachi, M. Amghar +3 more

We present a unified geometric and dynamical framework for a physical system consisting of $n$ spin-$1/2$ particles with all-range Ising interaction. Using the Fubini-Study formalism, we derive the metric tensor of the associated quantum state manifold and compute the corresponding Riemann curvature. Our analysis reveals that the system evolves over a smooth, compact, two-dimensional manifold with spherical topology and a dumbbell-like structure shaped by collective spin interactions. We further investigate the influence of the geometry and topology of the resulting state space on the behavior of geometric and topological phases acquired by the system. We explore how this curvature constrains the system's dynamical behavior, including its evolution speed and Fubini-Study distance between the quantum states. Within this geometric framework, we address the quantum brachistochrone problem and derive the minimal time required for optimal evolution, a result useful for time-efficient quantum circuit design. Subsequently, we explore the role of entanglement in shaping the state space geometry, modulating geometric phase, and controlling evolution speed and brachistochrone time. Our results reveal that entanglement enhances dynamics up to a critical threshold, beyond which geometric constraints begin to hinder evolution. Moreover, entanglement induces critical shifts in the geometric phase, making it a sensitive indicator of entanglement levels and a practical tool for steering quantum evolution. This approach offers valuable guidance for developing quantum technologies that require time-efficient control strategies rooted in the geometry of quantum state space.

quant-pharXiv:2512.21400•Scientific Reports•Dec 26, 2025•0 citations

Geometry and speed of evolution for a system of arbitrary spins with long-range Ising-type interaction.

Yuri Krynytskyi, Andrij Romanovych Kuzmak

We study the evolution of N arbitrary spins described by the long-range Ising-type interaction. This fact allows us to obtain the Fubiny-Study metric of the manifold defined by this evolution. We explore the topology of this manifold and shown that it corresponds to a sphere. With the help of this fact the manifold geometry is determined. Also we calculate the Riemann curvature of manifold and represent it by means of the speed of evolution. This is important for an experimental measurement of the curvature. The conditions for achieving the minimal and maximal values of the speed of evolution are obtained. Finally, we examine the influence of an external magnetic field on the metric of manifold and on the speed of evolution. We obtain the conditions for achieving the minimal possible speed of evolution. For some predefined initial states the orientations of magnetic fields for reaching the minimal and maximal values of the speed are found.

•May 25, 2018•0 citations

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

Jamal Elfakir

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-pharXiv:2605.00241•May 4, 2026•0 citations
Abstract We study the quantum evolution of a two-spin system described by the isotropic Heisenberg Hamiltonian in the external magnetic field. It is shown that this evolution happens on a two-parametric closed manifold. The Fubini–Study metric of this manifold is obtained. It is found that this is the metric of the torus. The entanglement of the states which belong to this manifold is investigated.
We consider a physical system consisting of two interacting spins governed by the [Formula: see text]-type Heisenberg Hamiltonian in an external magnetic field. We investigate the quantum evolution and the Riemannian geometry of the two-spin state space by means of the relevant Fubini–Study metric. The geometrical phase accumulated by the two-spin state is also examined under arbitrary and cyclic evolutions. By computing the evolution speed and the corresponding geodesic distance, we solve the quantum brachistochrone problem. The entanglement between the two spins is also studied via two approaches: the first one deals with the entanglement effect on the Fubini–Study metric and the geometrical phase, while the second one treats the entanglement effect on the evolution speed and the corresponding geodesic distance. Finally, we solve the quantum brachistochrone problem using the entanglement degree.
We consider a two-spin system of [Formula: see text] Heisenberg type submitted to an external magnetic field. Using the associated [Formula: see text] geometry, we investigate the dynamics of the system. We explicitly give the corresponding Fubini–Study metric. We show that for arbitrary pure initial states, the dynamics occurs on a torus. We compute the geometric phase, the dynamic phase and the topological phase. We investigate the interplay between the torus geometry and the entanglement of the two spins. In this respect, we provide a detailed analysis of the geometric phase, the dynamics velocity and the geodesic distance measured by the Fubini–Study metric in terms of the degree of entanglement between the two spins.
We provide explicit geometric description of state manifolds obtained from evolution governed by a four-parameter family of time-independent Hamiltonians. We cover most cases related to the real interacting two-qubit systems and discuss possible types of evolutions in terms of the defining parameters. The relevant description of the pure state spaces and their Riemannian geometry with the Fubini–Study metric is given. In particular, we analyze the modification of known geometry of quantum state manifold by the linear noncommuting perturbation of the Hamiltonian. Finally, we investigate the behavior of the entanglement for obtained families of states resulting from the unitary evolution.
We present a unified geometric and dynamical framework for a physical system consisting of nnn spin-1/21/21/2 particles with all-range Ising interaction. Using the Fubini-Study formalism, we derive the metric tensor of the associated quantum state manifold and compute the corresponding Riemann curvature. Our analysis reveals that the system evolves over a smooth, compact, two-dimensional manifold with spherical topology and a dumbbell-like structure shaped by collective spin interactions. We further investigate the influence of the geometry and topology of the resulting state space on the behavior of geometric and topological phases acquired by the system. We explore how this curvature constrains the system's dynamical behavior, including its evolution speed and Fubini-Study distance between the quantum states. Within this geometric framework, we address the quantum brachistochrone problem and derive the minimal time required for optimal evolution, a result useful for time-efficient quantum circuit design. Subsequently, we explore the role of entanglement in shaping the state space geometry, modulating geometric phase, and controlling evolution speed and brachistochrone time. Our results reveal that entanglement enhances dynamics up to a critical threshold, beyond which geometric constraints begin to hinder evolution. Moreover, entanglement induces critical shifts in the geometric phase, making it a sensitive indicator of entanglement levels and a practical tool for steering quantum evolution. This approach offers valuable guidance for developing quantum technologies that require time-efficient control strategies rooted in the geometry of quantum state space.
We study the evolution of N arbitrary spins described by the long-range Ising-type interaction. This fact allows us to obtain the Fubiny-Study metric of the manifold defined by this evolution. We explore the topology of this manifold and shown that it corresponds to a sphere. With the help of this fact the manifold geometry is determined. Also we calculate the Riemann curvature of manifold and represent it by means of the speed of evolution. This is important for an experimental measurement of the curvature. The conditions for achieving the minimal and maximal values of the speed of evolution are obtained. Finally, we examine the influence of an external magnetic field on the metric of manifold and on the speed of evolution. We obtain the conditions for achieving the minimal possible speed of evolution. For some predefined initial states the orientations of magnetic fields for reaching the minimal and maximal values of the speed are found.
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.