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

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 $PS{L}_2(7)$ subgroup of the automorphism group of the hexagon is discussed in detail and the Coxeter subgeometry is found to be intricately related to the $E_7$-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.

Figure 1

The Fano plane and its dual as the fundamental building blocks of the three-qubit Pauli group algebra.

Figure 2

The algebra of the operators from the two distinguished sets in terms of the incidence graph of the Fano plane.

Figure 3

The oriented dual Fano plane.

Figure 4

A diagrammatic illustration of the structure of the split Cayley hexagon of order two (after 20, 21, 22) and how this geometry grasps the core features of the algebra of the real three-qubit Pauli matrices by bijectively associating the latter with the points of the hexagon. The points themselves are represented by circles whose interior is colored (in an online version only) in nine different ways reflecting the nine orbits of an automorphism of order seven; the lines are drawn as black, light, and dark gray curves/joints. The remaining symbols and notation are explained in the text.

Figure 5

The Coxeter graph, in a form showing its automorphism of order seven, as a subgraph/subgeometry of the hexagon (see Fig. 4).