Geometric entanglement of symmetric states and the majorana representation

Authors:
Martin Aulbach;Damian Markham;Mio Murao
Affiliations:
The School of Physics and Astronomy, University of Leeds, Leeds, United Kingdom and Department of Physics, University of Oxford, Clarendon Laboratory, Oxford, United Kingdom;CNRS, LTCI, Telecom ParisTech, Paris, France;Department of Physics, Graduate School of Science, The University of Tokyo, Tokyo, Japan and Institute for Nano Quantum Information Electronics, The University of Tokyo, Tokyo, Japan
Venue:
TQC'10 Proceedings of the 5th conference on Theory of quantum computation, communication, and cryptography
Year:
2010

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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 (2010) 073025], 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.