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arXiv:1709.07067 · Journal of Physics A Mathematical and Theoretical

Entanglement and quantum state geometry of a spin system with all-range Ising-type interaction

A R Kuzmak

Published Mar 15, 2018• 15 citations
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Abstract

Abstract The evolution of an N spin-1/2 system with all-range Ising-type interaction is considered. For this system we study the entanglement of one spin with the rest spins. It is shown that the entanglement depends on the number of spins and the initial state. Also, the geometry of the manifold, which contains entangled states, is obtained. For this case we find the dependence of entanglement on the scalar curvature of the manifold and examine it for different numbers of spins in the system. Finally we show that the transverse magnetic field leads to a change in the manifold topology.

DOI
10.1088/1751-8121/aab6f8

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Abstract The evolution of an N spin-1/2 system with all-range Ising-type interaction is considered. For this system we study the entanglement of one spin with the rest spins. It is shown that the entanglement depends on the number of spins and the initial state. Also, the geometry of the manifold, which contains entangled states, is obtained. For this case we find the dependence of entanglement on the scalar curvature of the manifold and examine it for different numbers of spins in the system. Finally we show that the transverse magnetic field leads to a change in the manifold topology.
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.
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.
Introduction: In this study, we investigate the dynamics of the Entanglement in a two-spin system with long-range interaction.Method: For this purpose, we use negativity as the entanglement measurement. Using the time evolution operator, we obtain entanglement dynamics of the system at time t. We consider two different initial states and investigate the dynamical behavior of the system for all of them separately. Finally, we consider both J and D as a function of R and then study the time evolution of the entanglement in different R.Results: We find that in the long-range interaction, the R dependence of the dynamical behavior in the two systems is different. Depending on the initial states, the DM interaction and the magnetic field have no effect on the entanglement dynamics.
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 N\mathcal{N}N interacting spin-s\mathtt{s}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-s\mathtt{s}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.
We consider N initially disentangled spins, embedded in a ring or d-dimensional lattice of arbitrary geometry, which interact via some long-range Ising-type interaction. We investigate relations between entanglement properties of the resulting states and the distance dependence of the interaction in the limit N-->infinity. We provide a sufficient condition when bipartite entanglement between blocks of L neighboring spins and the remaining system saturates and determine S(L) analytically for special configurations. We find an unbounded increase of S(L) as well as diverging correlation and entanglement length under certain circumstances. For arbitrarily large N, we can efficiently calculate all quantities associated with reduced density operators of up to ten particles.
Abstract We consider one dimensional Ising spin system in a transverse uniform time-dependent magnetic field. The asymptotic behavior of the bipartite entanglements between the terminal spin and each one of the other spins along the chain is investigated and compared at different spin-spin interaction ranges, from nearest neighbor to infinite long range, under the separate action of two different magnetic fields, constant and time-varying. We find that each of the nearest neighbor and next to nearest neighbor bipartite entanglements reach an asymptotic final state that is independent of the initial condition or the variation in the interaction range showing perfect ergodic behavior at quite short interaction ranges. However, the nearest neighbor entanglement maintains this behavior at a slightly longer ranges. The other bipartite entanglements assume a zero value within these interaction ranges. At intermediate short and long interaction ranges, the asymptotic states of all entanglements become strongly dependent on the initial state and the interaction range, deviating from the ergodic behavior observed before. The maximum asymptotic entanglement attainable between a pair of spins takes place at a long interaction range value that increases with the distance between the spins. At the infinite long range interaction, the dynamics of all bipartite entanglements coincide. great care should be taken in constructing both.