Entanglement and symmetry in permutation symmetric states

TL;DR

This paper explores how permutation symmetry in multipartite quantum states relates to entanglement properties, using geometric and symmetry-based methods to classify entanglement types.

Contribution

It introduces a geometric framework linking state symmetry to entanglement measures and classifies entanglement types via symmetry considerations.

Findings

Symmetry under rotation corresponds to local unitary symmetry.

Geometric measure of entanglement is formulated as an optimization problem.

Different symmetries relate to distinct SLOCC entanglement classes.

Abstract

We investigate the relationship between multipartite entanglement and symmetry, focusing on permutation symmetric states. We use the Majorana representation, where these states correspond to points on a sphere. Symmetry of the representation under rotation is equivalent to symmetry of the states under products of local unitaries. The geometric measure of entanglement is thus phrased entirely as a geometric optimisation, and a condition for the equivalence of entanglement measures written in terms of point symmetries. Finally we see that different symmetries of the states correspond to different types of entanglement with respect to SLOCC interconvertibility.