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

TL;DR

This paper explores the deep connection between the automorphisms of the split Cayley hexagon and the structure of three-qubit magic configurations, revealing a surprising numerical coincidence and linking geometric hyperplanes to quantum contextuality.

Contribution

It uncovers a novel relationship between the automorphisms of the split Cayley hexagon and three-qubit magic configurations, suggesting a geometric underpinning of quantum contextuality.

Findings

12096 automorphisms of the split Cayley hexagon relate to 12096 magic pentagrams

Geometric hyperplanes of the hexagon are linked to magic configurations

Entanglement properties of configurations are analyzed

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.