Three-qubit operators, the split Cayley hexagon of order two, and black holes

P. Lévay, Metod Saniga, Péter Vrana

Research output: Contribution to journalArticle

47 Citations (Scopus)

Abstract

The set of 63 real generalized Pauli matrices of three-qubits can be factored into two subsets of 35 symmetric and 28 antisymmetric elements. This splitting is shown to be completely embodied in the properties of the Fano plane; the elements of the former set being in a bijective correspondence with the 7 points, 7 lines, and 21 flags, whereas those of the latter set having their counterparts in 28 antiflags of the plane. This representation naturally extends to the one in terms of the split Cayley hexagon of order two. Sixty three points of the hexagon split into 9 orbits of 7 points (operators) each under the action of an automorphism of order 7. Sixty three lines of the hexagon carry three points each and represent the triples of operators such that the product of any two gives, up to a sign, the third one. Since this hexagon admits a full embedding in a projective 5-space over GF(2), the 35 symmetric operators are also found to answer to the points of a Klein quadric in such space. The 28 antisymmetric matrices can be associated with the 28 vertices of the Coxeter graph, one of two distinguished subgraphs of the hexagon. The PSL2(7) subgroup of the automorphism group of the hexagon is discussed in detail and the Coxeter subgeometry is found to be intricately related to the E7-symmetric black-hole entropy formula in string theory. It is also conjectured that the full geometry/symmetry of the hexagon should manifest itself in the corresponding black-hole solutions. Finally, an intriguing analogy with the case of Hopf sphere fibrations and a link with coding theory are briefly mentioned.

Original languageEnglish
Article number124022
JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
Volume78
Issue number12
DOIs
Publication statusPublished - Dec 2 2008

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hexagons
operators
matrices
subgroups
string theory
embedding
set theory
coding
apexes
entropy
orbits
symmetry
products
geometry

ASJC Scopus subject areas

  • Nuclear and High Energy Physics

Cite this

Three-qubit operators, the split Cayley hexagon of order two, and black holes. / Lévay, P.; Saniga, Metod; Vrana, Péter.

In: Physical Review D - Particles, Fields, Gravitation and Cosmology, Vol. 78, No. 12, 124022, 02.12.2008.

Research output: Contribution to journalArticle

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