Quantum contextual finite geometries from dessins d'enfants

Abstract

We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the field $\bar{\mathbb{Q}}$ of algebraic numbers --- so-called Grothendieck's {\it dessins d'enfants} --- and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some 'exotic' geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two- and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of closely related graphs, namely the Schläfli and Clebsch ones. These findings seem to indicate that {\it dessins d'enfants} may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality.

18 pages

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Cited by in corpus (8)

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• Group geometrical axioms for magic states of quantum computing
• Finite Mathematics, Finite Quantum Theory And A Conjecture On The Nature Of Time
• Two-letter words and a fundamental homomorphism ruling geometric contextuality
• Geometric contextuality from the Maclachlan-Martin Kleinian groups
• A moonshine dialogue in mathematical physics