Geometrical entanglement of highly symmetric multipartite states and the Schmidt decomposition

TL;DR

This paper analyzes a geometric measure of entanglement for highly symmetric multipartite states, providing analytic solutions for the extrema of the distance function and linking these to the Schmidt decomposition.

Contribution

It offers a detailed analytical approach to compute entanglement measures for symmetric states using the Schmidt decomposition, extending previous geometric entanglement studies.

Findings

Analytic solutions for extrema of the distance function in symmetric states

Demonstration that solutions correspond to local minima

Connection between extremal solutions and 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.