The maximally entangled symmetric state in terms of the geometric measure

TL;DR

This paper investigates the maximum geometric entanglement of permutation symmetric states of qubits using the Majorana representation, combining analytical and numerical methods to identify highly entangled states up to 12 qubits.

Contribution

It introduces a novel approach to find maximally entangled symmetric states by exploiting symmetries and comparing the optimization problem with classical sphere problems.

Findings

Identified most entangled symmetric states up to 12 qubits.

Demonstrated how symmetries simplify entanglement calculations.

Compared the entanglement optimization with classical sphere problems.

Abstract

The geometric measure of entanglement is investigated for permutation symmetric pure states of multipartite qubit systems, in particular the question of maximum entanglement. This is done with the help of the Majorana representation, which maps an n qubit symmetric state to n points on the unit sphere. It is shown how symmetries of the point distribution can be exploited to simplify the calculation of entanglement and also help find the maximally entangled symmetric state. Using a combination of analytical and numerical results, the most entangled symmetric states for up to 12 qubits are explored and discussed. The optimization problem on the sphere presented here is then compared with two classical optimization problems on the S^2 sphere, namely Toth's problem and Thomson's problem, and it is observed that, in general, they are different problems.