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Erschienen in:
01.07.2015
Geometry of contextuality from Grothendieck’s coset space
Erschienen in: Quantum Information Processing | Ausgabe 7/2015
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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.