Geometric Entanglement of Symmetric States and the Majorana Representation

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.

10 pages, 8 figures; submitted to Lecture Notes in Computer Science (LNCS)

References in corpus (13)

• Experimental entanglement of a six-photon symmetric Dicke state
• Universal resources for measurement-based quantum computation
• Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model
• Entanglement and permutational symmetry
• The geometric measure of entanglement for symmetric states
• Multiqubit symmetric states with high geometric entanglement
• Entanglement of multiparty stabilizer, symmetric, and antisymmetric states
• Optimal state for keeping reference frames aligned and the Platonic solids
• Classifying Vortices in S=3 Bose-Einstein Condensates
• Maximally entangled three-qubit states via geometric measure of entanglement
• Inert states of spin-S systems
• Connecting the generalized robustness and the geometric measure of entanglement
• Reduced State Uniquely Defines Groverian Measure of Original Pure State

Cited by in corpus (5)

• Entanglement in the symmetric sector of $n$ qubits
• Nonlocality and Entanglement for Symmetric States
• Geometric description of modular and weak values in discrete quantum systems using the Majorana representation
• Anticoherent Subspaces
• Non-Abelian Berry transport, spin coherent states, and Majorana points