Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon

Abstract

Disregarding the identity, the remaining 63 elements of the generalized three-qubit Pauli group are found to contain 12096 distinct copies of Mermin's magic pentagram. Remarkably, 12096 is also the number of automorphisms of the smallest split Cayley hexagon. We give a few solid arguments showing that this may not be a mere coincidence. These arguments are mainly tied to the structure of certain types of geometric hyperplanes of the hexagon. It is further demonstrated that also an (18_{2}, 12_{3})-type of magic configurations, recently proposed by Waegell and Aravind (J. Phys. A: Math. Theor. 45 (2012) 405301), seems to be intricately linked with automorphisms of the hexagon. Finally, the entanglement properties exhibited by edges of both pentagrams and these particular Waegell-Aravind configurations are addressed.

15 pages, 4 figures, 5 tables

References in corpus (5)

• Hidden Variables and the Two Theorems of John Bell
• Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes
• On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits
• Geometric descriptions of entangled states by auxiliaries varieties
• 'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon

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• Geometric contextuality from the Maclachlan-Martin Kleinian groups
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• A new heuristic approach for contextuality degree estimates and its four- to six-qubit portrayals
• Three-Qubit-Embedded Split Cayley Hexagon is Contextuality Sensitive
• Veldkamp Spaces: From (Dynkin) Diagrams to (Pauli) Groups