Geometric Entanglement of Symmetric States and the Majorana Representation

摘要

Permutation-symmetric quantum states appear in a variety of physical situations, and they have been proposed for quantum infor mation tasks. This article builds upon the results of [New J. Phys. 12 (2010) 073025], where the maximally entangled symmetric states of up to twelve qubits were explored, and their amount of geometric entangle ment 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 Pla tonic 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 sym metric states as resources for measurement-based quantum computing is also discussed.