# 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

, Volume 14 (7) – May 5, 2015
13 pages

/lp/springer_journal/geometry-of-contextuality-from-grothendieck-s-coset-space-367YDf0p5e
Publisher
Springer US
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

### References

• 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
that matters to you.

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. ### 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$49/month

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.

$588$360/year

billed annually

14-day Free Trial