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. Quantum Information Processing Springer Journals

Geometry of contextuality from Grothendieck’s coset space

Loading next page...
Springer US
Copyright © 2015 by Springer Science+Business Media New York
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
Publisher site
See Article on Publisher Site


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

You’re reading a free preview. Subscribe to read the entire article.

DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches


Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.



billed annually
Start Free Trial

14-day Free Trial