Entanglement in the symmetric sector of $n$ qubits

TL;DR

This paper explores the entanglement properties of symmetric $n$-qubit states using the Majorana representation, identifying invariants under LU and SLOCC transformations and analyzing maximally entangled states.

Contribution

It introduces a geometric approach to entanglement invariants for symmetric states using Majorana points and cross ratios, extending understanding for three and four qubits.

Findings

Majorana representation maps symmetric states to points on a sphere.

Invariants under LU are constructed from inner products of Majorana vectors.

Cross ratios serve as invariants under SLOCC for four qubits.

Abstract

We discuss the entanglement properties of symmetric states of $n$ qubits. The Majorana representation maps a generic such state into a system of $n$ points on a sphere. Entanglement invariants, either under local unitaries (LU) or stochastic local operations and classical communication (SLOCC), can then be addressed in terms of the relative positions of the Majorana points. In the LU case, an over complete set of invariants can be built from the inner product of the radial vectors pointing to these points; this is detailed for the well documented three-qubits case. In the SLOCC case, cross ratio of related M^bius transformations are shown to play a central role, examplified here for four qubits. Finally, as a side result, we also analyze the manifold of maximally entangled 3 qubit state, both in the symmetric and generic case.