Geometry of a two-spin quantum state in evolution

TL;DR

This paper explores the geometric structure of a two-spin quantum system's evolution, revealing it occurs on a torus-shaped manifold and analyzing the entanglement properties within this geometric framework.

Contribution

It characterizes the evolution manifold of a two-spin system under the Heisenberg Hamiltonian and derives its Fubini-Study metric as a torus, linking geometry with quantum entanglement.

Findings

Evolution occurs on a two-parametric torus manifold.

Fubini-Study metric of the manifold is explicitly derived.

Entanglement properties are analyzed within this geometric setting.

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.