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- Title
Geometry of contextuality from Grothendieck's coset space.
- Authors
Planat, Michel
- Abstract
The geometry of cosets in the subgroups $$H$$ of the two-generator free group $$G=\langle a,b\rangle $$ 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 $$[\mathcal {A},\mathcal {B}] = \mathcal {A}\mathcal {B}-\mathcal {B}\mathcal {A}$$ 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.
- Subjects
GROTHENDIECK groups; DESSINS d'enfants (Mathematics); SCIENTIFIC observation; COMMUTATORS (Operator theory); QUANTUM theory
- Publication
Quantum Information Processing, 2015, Vol 14, Issue 7, p2563
- ISSN
1570-0755
- Publication type
Article
- DOI
10.1007/s11128-015-1004-2