The geometric measure of entanglement for symmetric states

TL;DR

This paper proves that for symmetric multiparticle quantum states, the closest product state in terms of the geometric measure of entanglement can always be chosen to be symmetric, simplifying entanglement quantification.

Contribution

It establishes that the closest product state to any symmetric multiparticle state is necessarily symmetric, advancing understanding of entanglement measures in symmetric quantum systems.

Findings

Closest product states can be chosen symmetric for symmetric states

Symmetric states have symmetric closest product states

Results apply to translationally invariant states in spin models

Abstract

Is the closest product state to a symmetric entangled multiparticle state also symmetric? This question has appeared in the recent literature concerning the geometric measure of entanglement. First, we show that a positive answer can be derived from results concerning symmetric multilinear forms and homogeneous polynomials, implying that the closest product state can be chosen to be symmetric. We then prove the stronger result that the closest product state to any symmetric multiparticle quantum state is necessarily symmetric. Moreover, we discuss generalizations of our result and the case of translationally invariant states, which can occur in spin models.