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

[摘要] 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 V 22 (37;0,12,15,10) and 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.

[关键词] ‘magic’ configurations of observables;three-qubit Pauli group;split Cayley hexagon of order two