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

TL;DR

This paper explores the geometric, topological, and dynamical properties of a long-range Ising model with interacting spins, linking these structures to quantum entanglement and solving related optimization problems.

Contribution

It introduces a comprehensive geometric analysis of the system, including phase space, curvature, and entanglement interplay, and addresses the quantum brachistochrone problem in this context.

Findings

Dynamics occur on a spherical topology manifold.

Entanglement influences evolution speed and geodesic distance.

The quantum brachistochrone problem is solved considering entanglement.

Abstract

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.