Hexagons govern three-qubit contextuality

Metod Saniga; Frédéric Holweck; Colm Kelleher; Axel Muller; Alain Giorgetti; Henri de Boutray

doi:10.22331/q-2025-01-20-1601

Hexagons govern three-qubit contextuality

Metod Saniga¹, Frédéric Holweck^2,3, Colm Kelleher², Axel Muller⁴, Alain Giorgetti⁴, and Henri de Boutray⁵

¹Astronomical Institute of the Slovak Academy of Sciences, SK-05960 Tatranská Lomnica, Slovakia
²Laboratoire Interdisciplinaire Carnot de Bourgogne, ICB/UTBM, UMR 6303, CNRS, Université de Technologie de Belfort-Montbéliard, F-90010 Belfort Cedex, France
³Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA
⁴Université Marie et Louis Pasteur, CNRS, Institut FEMTO-ST, F-25000 Besançon, France
⁵ColibriTD, F-75013 Paris, France

Published:	2025-01-20, volume 9, page 1601
Editor:	Ion Nechita
Eprint:	arXiv:2312.07738v3
Doi:	https://doi.org/10.22331/q-2025-01-20-1601
Citation:	Quantum 9, 1601 (2025).

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Updated after initial publication: This publication was updated to version v3 after the initial publication. The authors left the following comment on the arXiv:

28 pages, 15 figures, published in Quantum

Abstract

Split Cayley hexagons of order two are distinguished finite geometries living in the three-qubit symplectic polar space in two different forms, called classical and skew. Although neither of the two yields observable-based contextual configurations of their own, $classically$-embedded copies are found to fully encode contextuality properties of the most prominent three-qubit contextual configurations in the following sense: for each set of unsatisfiable contexts of such a contextual configuration there exists some classically-embedded hexagon sharing with the configuration exactly this set of contexts and nothing else. We demonstrate this fascinating property first on the configuration comprising all 315 contexts of the space and then on doilies, both types of quadrics as well as on complements of skew-embedded hexagons. In connection with the last-mentioned case and elliptic quadrics we also conducted some experimental tests on a Noisy Intermediate Scale Quantum (NISQ) computer to substantiate our theoretical findings.

Featured image: An illustration of the procedure that shows that to each ‘non-planar’ line (black) of a skew-embedded split Cayley hexagon of order two (left) one can associate a unique linear doily (right) that shares with it the maximum possible number of lines.

Popular summary

The paper has a specific look at observable-based proofs of quantum contextuality in the three-qubit symplectic polar space. Given certain sets of quantum contextual configurations in this space, we analyze arrangements of their (three-element) contexts that cannot be satisfied by an Non Contextual Hidden Variables (NCHV) model. For the most distinguished contextual configurations of this space the corresponding arrangements of unsatisfiable contexts exhibit a truly remarkable property, being nothing but proper intersections with particular copies of another dominant geometry of this space, called the smallest split Cayley hexagon. In the limiting case entailing all the 315 contexts of the space, the associated unsatisfiable part is found to cover the whole hexagon.

► BibTeX data

@article{Saniga2025hexagonsgovernthree,
  doi = {10.22331/q-2025-01-20-1601},
  url = {https://doi.org/10.22331/q-2025-01-20-1601},
  title = {Hexagons govern three-qubit contextuality},
  author = {Saniga, Metod and Holweck, Fr{\'{e}}d{\'{e}}ric and Kelleher, Colm and Muller, Axel and Giorgetti, Alain and Boutray, Henri de},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1601},
  month = jan,
  year = {2025}
}

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Cited by

[1] Axel Muller, Metod Saniga, Alain Giorgetti, Frédéric Holweck, and Colm Kelleher, "A new heuristic approach for contextuality degree estimates and its four- to six-qubit portrayals", Journal of Physics A: Mathematical and Theoretical 58 21, 215302 (2025).

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