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

TL;DR

This work develops a geometric framework for an $n$-spin-$ frac{1}{2}$ system with all-range Ising interactions, deriving the Fubini–Study metric and Riemann curvature of the quantum state manifold. The authors show the evolution occurs on a bounded two-dimensional manifold with spherical topology and a dumbbell shape, and they connect geometry to dynamical features such as geometric phases, evolution speed, and the quantum brachistochrone. By analyzing both the full $n$-spin system and a two-spin reduction, they reveal how entanglement reshapes state-space geometry, alters curvatures, and induces nonlinear, path-dependent geometric phases, while also unveiling a nonmonotonic speed–entanglement trade-off relevant for time-efficient quantum control. These insights offer practical guidance for designing fast, geometry-informed quantum protocols and holonomic gates in many-body spin systems.

Abstract

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.

Figures (7)

• Figure 1: Illustrative scheme of two coupled $n$-spin-$1/2$ chains under all-range Ising interaction.
• Figure 2: (a) Illustrative scheme of the quantum state space \ref{['f']} of the $n$ spin-1/2 system under the all-range Ising model. (b) the behavior of the state space curvature \ref{['h']} against the initial parameter $\theta$ for specific spin$-1/2$ numbers.
• Figure 3: The cyclic geometric phase \ref{['x1']} against the variable $\theta$ for specific spin$-1/2$ numbers.
• Figure 4: The evolution rate (speed) [Eq. \ref{['a7']}, panel (a)] and the FS-distance [Eq. \ref{['a8']}, panel (b)] versus the parameter $\theta$ for certain spin$-1/2$ numbers by setting $J=1$.
• Figure 5: The geometric measure of entanglement \ref{['a13']} versus the dynamic parameter $\chi$ for some values of $\theta$.