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 fieldQ of algebraic numbers -so-called Grothendieck's 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 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.

Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation

“…Following the impetus given by Grothendieck [16], it is now known that there are various ways to dress a group P generated by two permutations: (i) as a connected graph drawn on a compact oriented two-dimensional surface, a bicolored map (or hypermap) with n edges, B black points, W white points, F faces, genus g and Euler characteristic 2 − 2g = B + W + F − n [17]; (ii) as a Riemann surface X of the same genus equipped with a meromorphic function f from X to the Riemann sphereC unramified outside the critical set {0, 1, ∞}, the pair (X, f ) called a Belyi pair, and in this context, hypermaps are called dessins d'enfants [14,16]; (iii) as a subgroup H of the free group G = a, b (or of a two-generator group G = a, b|rels ) where P encodes the action of (right) cosets of H on the two generators a and b; the Coxeter-Todd algorithm does the job [11]; and finally (iv), when P is of rank at least three, that is of a point stabilizer with at least three orbits, as a non-trivial finite geometry [10][11][12][13]. Finite simple groups are generated by two of their elements [18], so that it is useful to characterize them as members of the categories just described.…”

“…We first show how to recover the geometry of the well-known Mermin square, a (3 × 3) grid, that is the basic model of two-qubit contextuality [23] (see Figure 7 in [10] and Figure 3i in [11]). Starting with group G = a, b|b 2 and making use of a mathematical software, such as Magma, one derives the (unique) subgroup H of G that is of index nine and possesses a permutation representation P isomorphic to the finite group P 36 = Z 2 3 × Z 2 2 reflecting the symmetry of the grid.…”

“…Starting with group G = a, b|b 2 and making use of a mathematical software, such as Magma, one derives the (unique) subgroup H of G that is of index nine and possesses a permutation representation P isomorphic to the finite group P 36 = Z 2 3 × Z 2 2 reflecting the symmetry of the grid. The permutation representation is as follows: Figure 1i) [10,11]. Next, we apply the procedure described in Item (iv) of Section 2.1.…”

“…Only recently, the first author observed that much of the information needed is encapsulated in permutation representations, of rank larger than two, available in the Atlas of finite group representations [9]. The coset enumeration methodology of the Atlas was used by us for deriving many finite geometries underlying quantum commutation and the related contextuality [10][11][12][13]. As a bonus, the two-generator permutation groups and their underlying geometries may luckily be considered as dessins d'enfants [14], although this topological and algebraic aspect of the finite simple (or not simple) groups is barely mentioned in the literature.…”

“…Another leg D signs the coset structure of the used subgroup H of the two-generator free group G (or of a quotient group G of G with relations), whose finite index [G, H] = n is the number of edges of D, and at the same time, the size of the set on which P acts, as in [11]. Finally, G is the geometry with n vertices that is defined/stabilized by D [10]. Then, in Section 3, we organize the relevant P − D − G tripods taken from the classes of the Atlas and find that many of them reflect quantum commutation, specifically the symplectic, unitary and orthogonal classes.…”