How to draw a hexagon

Schroth, Andreas E.

doi:10.1016/s0012-365x(98)00294-5

Cited by 19 publications

(16 citation statements)

References 4 publications

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“…It has 63 points and 63 lines; every point is contained in three lines and every line contains three points. One of its stunning pictures created by the method of "finite pottery" [18,19] is reproduced in Fig. 4.…”

Section: Core Geometry: the Smallest Split Cayley Hexagonmentioning

confidence: 99%

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

2008

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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 P SL 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.

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Section: Core Geometry: the Smallest Split Cayley Hexagonmentioning

confidence: 99%

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

2008

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“…In the latter work it was found that the E 8 -root system is the sought-for saturated three-qubit ray configuration. Here, we shall further advance the ideas of [9] and show that the finite geometry that seems to entail all essential features of three-qubit contextuality and associated magic configurations is indeed the split Cayley hexagon of order two (occasionally referred to as the G 2 (2)-hexagon since G 2 (2) is its automorphism group) [10,11].The paper is organized as follows. In Sec.…”

mentioning

confidence: 81%

“…A classical (left) and skew (right) symplectic embedding of the hexagon in terms of the elements of the three-qubit Pauli group. The points of the hexagon are represented by small circles and its lines by straight segments and/or arcs joining three circles each (based on the drawings given in [10,11] Table 1. A classification of the geometric hyperplanes of the hexagon.…”

Section: Three-qubit 'Magicity' and The Smallest Split Cayley Hexagonmentioning

confidence: 99%

Distinguished three-qubit ‘magicity’ via automorphisms of the split Cayley hexagon

Planat

Санига

Holweck

2013

Quantum Inf Process

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Abstract. Disregarding the identity, the remaining 63 elements of the generalized three-qubit Pauli group are found to contain 12 096 distinct copies of Mermin's magic pentagram. Remarkably, 12 096 is also the number of automorphisms of the smallest split Cayley hexagon. We give a few solid arguments showing that this may not be a mere coincidence. These arguments are mainly tied to the structure of certain types of geometric hyperplanes of the hexagon. It is further demonstrated that also an (18 2 , 12 3 )-type of magic configurations, recently proposed by Waegell and Aravind (J. Phys. A: Math. Theor. 45 (2012) 405301), seems to be intricately linked with automorphisms of the hexagon. Finally, the entanglement properties exhibited by edges of both pentagrams and these particular Waegell-Aravind configurations are addressed.

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“…A point of the four-qubit symplectic polar space, W(7, 2), is collinear with 126 other points of this space (see, for example, [4] and references therein). If this point lies off the Figure 1: A diagrammatic illustration of the structure of the split Cayley hexagon of order two (based on drawings given in [37,38]). The points are illustrated by small circles and its lines by triples of points lying on the same segments of straight-lines and/or arcs.…”

Section: Mermin's Pentagramsmentioning

confidence: 99%

Grassmannian connection between three- and four-qubit observables, Mermin’s contextuality and black holes

Lévay

Planat

Санига

2013

J. High Energ. Phys.

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We invoke some ideas from finite geometry to map bijectively 135 heptads of mutually commuting three-qubit observables into 135 symmetric four-qubit ones. After labeling the elements of the former set in terms of a seven-dimensional Clifford algebra, we present the bijective map and most pronounced actions of the associated symplectic group on both sets in explicit forms. This formalism is then employed to shed novel light on recentlydiscovered structural and cardinality properties of an aggregate of three-qubit Mermin's "magic" pentagrams. Moreover, some intriguing connections with the so-called black-holequbit correspondence are also pointed out.

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How to draw a hexagon

Cited by 19 publications

References 4 publications

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

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

Distinguished three-qubit ‘magicity’ via automorphisms of the split Cayley hexagon

Grassmannian connection between three- and four-qubit observables, Mermin’s contextuality and black holes

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