The maximally entangled symmetric state in terms of the geometric measure

Abstract

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