`Magic' configurations of three-qubit observables and geometric hyperplanes of the smallest split Cayley hexagon

Metod Saniga ; Michel Planat ; Petr Pracna ; Péter Lévay

Symmetry, integrability and geometry: methods and applications, Tome 8 (2012)

Résumé

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 $2_414_2-4_36_4$ ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types $\mathcal V_{22}(37;0,12,15,10)$ and $\mathcal V_4(49;0,0,21,28)$ 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.

Détail

Keywords: ‘magic’ configurations of observables; three-qubit Pauli group; split Cayley hexagon of order two.

Bibliographie

[1] De Kaey J., Van Maldeghem H., “A characterization of the split Cayley generalized hexagon $H(q)$ using one subhexagon of order $(1,q)$”, Discrete Math., 294 (2005), 109–118 | DOI | MR | Zbl

[2] Frohardt D., Johnson P., “Geometric hyperplanes in generalized hexagons of order $(2,2)$”, Comm. Algebra, 22 (1994), 773–797 | DOI | MR | Zbl

[3] Havlicek H., Odehnal B., Saniga M., “Factor-group-generated polar spaces and (multi-)qudits”, SIGMA, 5 (2009), 096, 15 pp. ; arXiv: 0903.5418 | DOI | MR | Zbl

[4] Kochen S., Specker E.P., “The problem of hidden variables in quantum mechanics”, J. Math. Mech., 17 (1967), 59–87 | MR | Zbl

[5] Lévay P., Saniga M., Vrana P., “Three-qubit operators, the split Cayley hexagon of order two, and black holes”, Phys. Rev. D, 78 (2008), 124022, 16 pp. ; arXiv: 0808.3849 | DOI

[6] Mermin N.D., “Hidden variables and the two theorems of John Bell”, Rev. Modern Phys., 65 (1993), 803–815 | DOI | MR

[7] Planat M., “Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function?”, J. Phys. A: Math. Theor., 44 (2011), 045301, 16 pp. ; arXiv: 1009.3858 | DOI | MR | Zbl

[8] Planat M., Saniga M., “On the Pauli graphs on $N$-qudits”, Quantum Inf. Comput., 8 (2008), 127–146 ; arXiv: quant-ph/0701211 | MR | Zbl

[9] Polster B., Schroth A.E., van Maldeghem H., “Generalized flatland”, Math. Intelligencer, 23 (2001), 33–47 | DOI | MR

[10] Ronan M.A., “Embeddings and hyperplanes of discrete geometries”, European J. Combin., 8 (1987), 179–185 | MR | Zbl

[11] Saniga M., Lévay P., “Mermin's pentagram as an ovoid of PG(3,2)”, Europhys. Lett., 97 (2012), 50006, 3 pp. ; arXiv: 1111.5923 | DOI

[12] Saniga M., Planat M., “Finite geometry behind the Harvey–Chryssanthacopoulos four-qubit magic rectangle”, Quantum Inf. Comput., 11 (2012), 1011–1016; arXiv: 1204.6229

[13] Saniga M., Planat M., “Multiple qubits as symplectic polar spaces of order two”, Adv. Stud. Theor. Phys., 1 (2007), 1–4 ; arXiv: quant-ph/0612179 | MR | Zbl

[14] Saniga M., Planat M., Prachna P., “Projective curves over a ring that includes two-qubits”, Theoret. and Math. Phys., 155 (2008), 905–913 ; arXiv: quant-ph/0611063 | DOI | MR | Zbl

[15] Saniga M., Planat M., Pracna P., Havlicek H., “The Veldkamp space of two-qubits”, SIGMA, 3 (2007), 075, 7 pp. ; arXiv: 0704.0495 | DOI | MR | Zbl

[16] Schroth A.E., “How to draw a hexagon”, Discrete Math., 199 (1999), 161–171 | DOI | MR | Zbl

[17] Thas K., “The geometry of generalized Pauli operators of $N$-qudit Hilbert space, and an application to MUBs”, Europhys. Lett., 86 (2009), 60005, 3 pp. | DOI

[18] Vrana P., Lévay P., “The Veldkamp space of multiple qubits”, J. Phys. A: Math. Theor., 43 (2010), 125303, 16 pp. ; arXiv: 0906.3655 | DOI | MR | Zbl

[19] Waegell M., Aravind P.K., “Proofs of the Kochen–Specker theorem based on a system of three qubits”, J. Phys. A: Math. Theor., 45 (2012), 405301, 13 pp. ; arXiv: 1205.5015 | DOI | Zbl