Quantum state geometry and entanglement of two spins with anisotropic interaction in evolution

TL;DR

This paper explores the geometric structure and entanglement properties of two-spin systems under anisotropic interactions, analyzing their evolution on specific manifolds and calculating the Fubini-Study metric for different Hamiltonians.

Contribution

It provides a detailed geometric analysis of two-spin quantum evolution with anisotropic interactions and links different Hamiltonians through unitary transformations.

Findings

Evolution occurs on manifolds with geometry depending on interaction ratios.

Fubini-Study metric of the evolution manifold is explicitly calculated.

Entanglement properties are examined within the geometric framework.

Abstract

Quantum evolution of a two-spin system with anisotropic Heisenberg Hamiltonian in the magnetic field is considered. We show that this evolution happens on some manifold with geometry depending on the ratio between the interaction couplings and on the initial state. The Fubini-Study metric of this manifold is calculated. The entanglement of the states belonging to this manifold is examined. Also we investigate similar problem for a two-spin system described by the Dzyaloshinsky-Moria Hamiltonian. The problem is solved using the fact that this Hamiltonian and the anisotropic Heisenberg Hamiltonian are linked by the unitary transformation.