Geometric Entanglement of Symmetric States and the Majorana Representation

TL;DR

This paper explores the geometric entanglement of symmetric quantum states using the Majorana representation, revealing new symmetry relationships and analyzing their potential for quantum computing applications.

Contribution

It demonstrates the duality between Platonic solids and Majorana representations, and clarifies the nature of maximally entangled symmetric states beyond spherical designs.

Findings

Duality of Platonic solids reflected in Majorana representation.

Maximally entangled symmetric states are not generally anticoherent or spherical designs.

Discussion on symmetric states as resources for measurement-based quantum computing.

Abstract

Permutation-symmetric quantum states appear in a variety of physical situations, and they have been proposed for quantum information tasks. This article builds upon the results of [New J. Phys. 12, 073025 (2010)], where the maximally entangled symmetric states of up to twelve qubits were explored, and their amount of geometric entanglement determined by numeric and analytic means. For this the Majorana representation, a generalization of the Bloch sphere representation, can be employed to represent symmetric n qubit states by n points on the surface of a unit sphere. Symmetries of this point distribution simplify the determination of the entanglement, and enable the study of quantum states in novel ways. Here it is shown that the duality relationship of Platonic solids has a counterpart in the Majorana representation, and that in general maximally entangled symmetric states neither correspond to anticoherent spin states nor to spherical designs. The usability of symmetric states as resources for measurement-based quantum computing is also discussed.