Geometry of contextuality from Grothendieck’s coset space

Abstract

The geometry of cosets in the subgroups H of the two-generator free group G = a, b nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. In previous work, it was established that dessins stabilize point-line geometries whose incidence structure reflects the commutation of (generalized) Pauli operators. Now we find that the nonexistence of a dessin for which the commutator (a, b) = a −1 b −1 ab precisely corresponds to the commutator of quantum observables [A, B] = AB − BA on all lines of the geometry is a signature of quantum contextuality. This occurs first at index |G:H | = 9 in Mermin's square and at index 10 in Mermin's pentagram, as expected. Commuting sets of n-qubit observables with n > 3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.

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.…”

“…Further, G is said to be contextual if at least one of its lines/edges corresponds to a set/pair of vertices encoded by non-commuting cosets [11]. A straightforward measure of contextuality can be taken as the ratio κ = E c /E between the number E c of lines/edges of G with non-commuting cosets and the whole number E of lines/edges of G. Of course, lines/edges passing through the identity coset e have commuting vertices, so that one always as κ < 1.…”

“…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.…”

“…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.…”

“…More cautiously, in Section 2 of the present paper, we briefly account for the group concepts involved in our approach by defining a tripod P − D − G. One leg P is a desired two-generator permutation representation of a finite group P [9]. 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].…”