### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 083, 9 pages      arXiv:1206.3436      https://doi.org/10.3842/SIGMA.2012.083

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

Metod Saniga a, Michel Planat b, Petr Pracna c and Péter Lévay d
a) Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovak Republic
b) Institut FEMTO-ST, CNRS, 32 Avenue de l'Observatoire, F-25044 Besançon Cedex, France
c) J. Heyrovský Institute of Physical Chemistry, v.v.i., Academy of Sciences of the Czech Republic, Dolejškova 3, CZ-182 23 Prague 8, Czech Republic
d) Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary

Received June 22, 2012, in final form November 02, 2012; Published online November 06, 2012

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 182−123 and 24142−4364 ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types V22(37;0,12,15,10) and V4(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.

Key words: 'magic' configurations of observables; three-qubit Pauli group; split Cayley hexagon of order two.

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