### 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 language | English |
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Article number | 124022 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 78 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 2 2008 |

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### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Physical Review D - Particles, Fields, Gravitation and Cosmology*,

*78*(12), [124022]. https://doi.org/10.1103/PhysRevD.78.124022

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

Research output: Contribution to journal › Article

*Physical Review D - Particles, Fields, Gravitation and Cosmology*, vol. 78, no. 12, 124022. https://doi.org/10.1103/PhysRevD.78.124022

}

TY - JOUR

T1 - Three-qubit operators, the split Cayley hexagon of order two, and black holes

AU - Lévay, P.

AU - Saniga, Metod

AU - Vrana, Péter

PY - 2008/12/2

Y1 - 2008/12/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=58949084955&partnerID=8YFLogxK

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U2 - 10.1103/PhysRevD.78.124022

DO - 10.1103/PhysRevD.78.124022

M3 - Article

AN - SCOPUS:58949084955

VL - 78

JO - Physical review D: Particles and fields

JF - Physical review D: Particles and fields

SN - 1550-7998

IS - 12

M1 - 124022

ER -