Drawing Quantum Contextuality with ‘Dessins d’enfants’

Abstract

Abstract.In the standard formulation of quantum mechanics, there exists an inherent feedback of the measurement setting on the elementary object under scrutiny. Thus one cannot assume that an 'element of reality' prexists to the measurement and, it is even more intriguing that unperformed/counterfactual observables enter the game. This is called quantum contextuality. Simple finite projective geometries are a good way to picture the commutation relations of quantum observables entering the context, at least for systems with two or three parties. In the essay, it is further discovered a mathematical mechanism for 'drawing' the contexts. The so-called 'dessins d'enfants' of the celebrated mathematician Alexandre Grothendieck feature group, graph, topological, geometric and algebraic properties of the quantum contexts that would otherwise have been 'hidden' in the apparent randomness of measurement outcomes.

Geometry of contextuality from Grothendieck’s coset space

“…Finally, the coset structure and its permutation representation (by a dessin d'enfant) provide a coordinatization to many point-line geometries occurring in the investigation of commutation of quantum observables [8][9][10]. Taking the permutation group P (it identifies a dessin D) corresponding to a subgroup H of G, one proceeds by first listing the m non-isomorphic subgroups S m stabilizing a pair of elements/cosets.…”

“…We recently found a mathematical scheme giving rise to the aforementioned geometries as well as many related ones [8][9][10]. Our work is based on Grothendieck's great insight about the relationship between algebra, geometry and topology, called by him dessins d'enfants [11].…”