Geometry of contextuality from Grothendieck’s coset space

Geometry of contextuality from Grothendieck’s coset space The geometry of cosets in the subgroups $$H$$ H of the two-generator free group $$G=\langle a,b\rangle $$ 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 $$(a,b)=a^{-1}b^{-1}ab$$ ( a , b ) = a - 1 b - 1 a b precisely corresponds to the commutator of quantum observables $$[\mathcal {A},\mathcal {B}] = \mathcal {A}\mathcal {B}-\mathcal {B}\mathcal {A}$$ [ A , B ] = A B - B A on all lines of the geometry is a signature of quantum contextuality. This occurs first at index $$|G$$ | G : $$H|=9$$ H | = 9 in Mermin’s square and at index $$10$$ 10 in Mermin’s pentagram, as expected. Commuting sets of $$n$$ n -qubit observables with $$n>3$$ n > 3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Geometry of contextuality from Grothendieck’s coset space

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Publisher
Springer US
Copyright
Copyright © 2015 by Springer Science+Business Media New York
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1004-2
Publisher site
See Article on Publisher Site

Abstract

The geometry of cosets in the subgroups $$H$$ H of the two-generator free group $$G=\langle a,b\rangle $$ 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 $$(a,b)=a^{-1}b^{-1}ab$$ ( a , b ) = a - 1 b - 1 a b precisely corresponds to the commutator of quantum observables $$[\mathcal {A},\mathcal {B}] = \mathcal {A}\mathcal {B}-\mathcal {B}\mathcal {A}$$ [ A , B ] = A B - B A on all lines of the geometry is a signature of quantum contextuality. This occurs first at index $$|G$$ | G : $$H|=9$$ H | = 9 in Mermin’s square and at index $$10$$ 10 in Mermin’s pentagram, as expected. Commuting sets of $$n$$ n -qubit observables with $$n>3$$ n > 3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.

Journal

Quantum Information ProcessingSpringer Journals

Published: May 5, 2015

References

  • Hidden variables and two theorems of John Bell
    Mermin, ND
  • Experimentally testable state-independent quantum contextuality
    Cabello, A

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