Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes

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 anti-flags of the plane. This representation naturally extends to the one in terms of the split Cayley hexagon of order two. 63 points of the hexagon split into 9 orbits of 7 points (operators) each under the action of an automorphism of order 7. 63 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 PSL_{2}(7) subgroup of the automorphism group of the hexagon is discussed in detail and the Coxeter sub-geometry 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.

21 pages, 5 figures, 2 tables; Version 2: an equation and a short paragraph added (Sec. 6), notation and nomenclature slightly polished

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Cited by in corpus (16)

• On the Black-Hole/Qubit Correspondence
• Black Hole Entropy and Finite Geometry
• Factor-Group-Generated Polar Spaces and (Multi-)Qudits
• Mermin pentagrams arising from Veldkamp lines for three qubits
• Geometric and Information-Theoretic Properties of the Hoggar Lines
• A finite geometric toy model of space-time as an error correcting code
• STU attractors from vanishing concurrence
• Character varieties and algebraic surfaces for the topology of quantum computing
• A Kind of Magic
• A Sequence of Qubit-Qudit Pauli Groups as a Nested Structure of Doilies
• 'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon
• Hexagons govern three-qubit contextuality
• Three-Qubit-Embedded Split Cayley Hexagon is Contextuality Sensitive
• Quantum circuit simulation of black hole evaporation and Maxwell demon interpretation
• A new heuristic approach for contextuality degree estimates and its four- to six-qubit portrayals
• Finite Geometries with Qubit Operators