'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon

TL;DR

This paper links specific three-qubit observable configurations proving the Kochen-Specker theorem to geometric hyperplanes of the split Cayley hexagon, revealing new symmetries and extensions in quantum contextuality structures.

Contribution

It demonstrates that two known three-qubit observable sets can be uniquely extended into geometric hyperplanes of the split Cayley hexagon, and finds additional configurations via automorphisms.

Findings

Two configurations extend into specific hyperplanes of the split Cayley hexagon.

Six new replicas of these configurations are generated using hexagon automorphisms.

The work connects quantum contextuality proofs with geometric structures in finite geometry.

Abstract

Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13 pages] have given a number of distinct sets of three-qubit observables, each furnishing a proof of the Kochen-Specker theorem. Here it is demonstrated that two of these sets/configurations, namely the $18_{2} - 12_{3}$ and $2_{4}14_{2} - 4_{3}6_{4}$ ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types ${\cal V}_{22}(37; 0, 12, 15, 10)$ and ${\cal V}_{4}(49; 0, 0, 21, 28)$ in the classification of Frohardt and Johnson [Comm. Algebra 22 (1994), 773-797]. Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained.