Geometrical entanglement of highly symmetric multipartite states and the Schmidt decomposition

Abstract

In a previous paper we examined a geometric measure of entanglement based on the minimum distance between the entangled target state of interest and the space of unnormalized product states. Here we present a detailed study of this entanglement measure for target states with a large degree of symmetry. We obtain analytic solutions for the extrema of the distance function and solve for the Hessian to show that, up to the action of trivial symmetries, the solutions correspond to local minima of the distance function. In addition, we show that the conditions that determine the extremal solutions for general target states can be obtained directly by parametrizing the product states via their Schmidt decomposition.

16 pages, references added and discussion expanded

References in corpus (8)

• Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model
• The geometric measure of entanglement for symmetric states
• Hierarchies of Geometric Entanglement
• Geometric Entanglement in a One-Dimensional Valence Bond Solid State
• An Experimentally accessible geometric measure for entanglement in $N$-qubit pure states
• Geometric Measure of Entanglement and Shared Quantum States
• Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation
• Geometric measures of entanglement and the Schmidt decomposition

Cited by in corpus (3)

• Genuine Multipartite Entanglement Trends in Gapless-gapped Transitions of Quantum Spin Systems
• Characterizing Genuine Multisite Entanglement in Isotropic Spin Lattices
• On the geometric measure of entanglement for pure states