Geometry of contextuality from Grothendieck's coset space

Abstract

The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli operators [Information 5, 209 (2014); 1310.4267 and 1404.6986 (quant-ph)]. Now we find that the non-existence 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 \textgreater{} 3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.

13 pages, Quant. Inf. Proc

References in corpus (11)

• Hidden Variables and the Two Theorems of John Bell
• Experimentally testable state-independent quantum contextuality
• The status of determinism in proofs of the impossibility of a noncontextual model of quantum theory
• Three-Qubit Operators, the Split Cayley Hexagon of Order Two and Black Holes
• On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits
• Quantum Contextuality with Stabilizer States
• Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon
• Parity proofs of the Kochen-Specker theorem based on the 120-cell
• Quantum contextual finite geometries from dessins d'enfants
• It from qubit: how to draw quantum contextuality
• An order-theoretic quantification of contextuality

Cited by in corpus (10)

• Magic informationally complete POVMs with permutations
• Contextuality with a Small Number of Observables
• Quantum contextual finite geometries from dessins d'enfants
• Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation
• The magic of universal quantum computing with permutations
• Group geometrical axioms for magic states of quantum computing
• Quantum computation and measurements from an exotic space-time R4
• Geometric contextuality from the Maclachlan-Martin Kleinian groups
• A moonshine dialogue in mathematical physics
• Two-letter words and a fundamental homomorphism ruling geometric contextuality