Quantum contextual finite geometries from dessins d'enfants. - Archive ouverte HAL Access content directly
Journal Articles International Journal of Geometric Methods in Modern Physics Year : 2015

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 Schl\"{a}fli 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.
Fichier principal
Vignette du fichier
ContextsDessinsFinal.pdf (473.09 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-00873461 , version 1 (15-10-2013)
hal-00873461 , version 2 (28-10-2013)
hal-00873461 , version 3 (04-03-2015)
hal-00873461 , version 4 (04-09-2015)

Identifiers

Cite

Michel Planat, Alain Giorgetti, Frédéric Holweck, Metod Saniga. Quantum contextual finite geometries from dessins d'enfants.. International Journal of Geometric Methods in Modern Physics, 2015, 12, pp.1550067. ⟨10.1142/S021988781550067X⟩. ⟨hal-00873461v4⟩
397 View
303 Download

Altmetric

Share

Gmail Facebook X LinkedIn More