bibkey: saniga2025hexagons
authors: Metod Saniga; Frédéric Holweck; Colm Kelleher; Axel Muller; Alain Giorgetti; Henri de Boutray
year: 2025
title: "Hexagons govern three-qubit contextuality"
doi: 10.22331/q-2025-01-20-1601
url: "https://arxiv.org/abs/2312.07738v3"
claim: "The paper shows that classically-embedded split Cayley hexagons of order two in the three-qubit symplectic polar space W(5,2) share with the whole space, with both types of quadrics and with doilies exactly as many lines as the degree of contextuality of these line configurations, and conjectures that this holds for any contextual sub-geometry of W(5,2) whose contexts are lines."
strata_touched:
  - D5/S3/Quantum/Contextuality/PerpHyperplaneHexagonContextualityRefutation
license: citation-only
triage: anchor

Saniga, Holweck, Kelleher, Muller, Giorgetti and de Boutray, Hexagons govern three-qubit contextuality

M. Saniga, F. Holweck, C. Kelleher, A. Muller, A. Giorgetti and H. de Boutray, Quantum 9 (2025) 1601, DOI 10.22331/q-2025-01-20-1601; arXiv:2312.07738v3. Quotations are from the arXiv v3 TeX source.

The points of $\mathcal{W}(5,2)$ are the 63 three-qubit Pauli observables other than the identity, its lines the 315 triples of pairwise commuting observables, and a line is negative when its three observables multiply to $-I$. The degree of contextuality of a configuration is “the minimal number of contexts that cannot be satisfied by any NCHV model”. The split Cayley hexagon of order two has 63 points and 63 lines, three points per line and three lines per point, and smallest polygons hexagons; in a classically-embedded copy every point is planar, its three lines lying in a plane of $\mathcal{W}(5,2)$. Section 4 ends:

The message from the above-given observations is more than obvious: the degree of contextuality of a contextual quantum geometry of the three-qubit $\mathcal{W}(5,2)$ whose contexts are i) all the 315 lines of the whole space $\mathcal{W}(5,2)$, ii) all the 45 lines of an elliptic quadric, iii) all the 105 lines of a hyperbolic quadric or iv) all the 15 lines of a doily is equal to the number of lines each of these geometries shares with a classically-embedded hexagon, with the understanding that each shared line corresponds to an unsatisfiable context. A natural generalization of these findings is that this should hold for any contextual sub-geometry of $\mathcal{W}(5,2)$ whose contexts are lines.

The following paragraph treats the complement of a skew-embedded hexagon as “a particular stance of this conjecture”.

Verified locator

Source: https://arxiv.org/abs/2312.07738v3

• DOI: https://doi.org/10.22331/q-2025-01-20-1601 (journal version; title, authors, journal, volume and article number from Crossref).
• URL: https://arxiv.org/abs/2312.07738v3. The e-print member main.tex has 1623 lines and md5 89c88a7b954134b5014683870a6c51c4. Locations:
  • the split Cayley hexagon and its embeddings into $\mathcal{W}(5,2)$, l. 289–300;
  • the degree of contextuality, l. 341–352;
  • planar points and the classical embedding, l. 477–479;
  • the conjecture, l. 735–741, and the skew-hexagon complement as its particular case, l. 742–744.