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arXiv:2608.25008

Kochen-Specker Configurations from Grids on Dual Quadrics

Giuseppe Favacchio

math.AGquant-phSubmitted Aug 25, 2026• Announced Aug 27, 2026• 0 citations
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Abstract

We develop a geometric framework for constructing and organizing Kochen--Specker configurations in real four-dimensional space. The construction uses finite grids on pairs of smooth quadrics related by Euclidean polarity. Their incidence geometry directly produces orthogonal measurement contexts and parity proofs of quantum contextuality, yielding infinite families of configurations and a geometric interpretation and extension of a previously known cyclic construction. For a distinguished subfamily, the same geometry admits a canonical completion determined by secant lines. The first two instances of this completion recover the exceptional root configurations of types $F_4$ and $H_4$, while the smallest case also recovers the Cabello configuration and its embedding in the Peres configuration. Thus several prominent four-dimensional contextual configurations, previously obtained from different constructions, arise from a single projective-geometric mechanism.

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We study the orthogonality graphs Gn of the Kochen–Specker configurations Zn,n(kn, kn) recently obtained by Favacchio from grids on dual quadrics: for every odd n ≥ 3 they consist of 2n2 rays in R4 and n2 orthogonal bases, and G3 is the 18-ray set of Cabello, Estebaranz and García-Alcaine. We determine all orthogonal pairs, not only those inside the bases: Gn is (2n + 1)-regular, and the “only if” direction rests on the fact that 2 cos(2πt/n) is a unit in the ring of algebraic integers for odd n and t ̸≡ 0, while 2 is not. Consequently the independence number is α(Gn) = (n2 − 1)/2, with an explicit maximum independent set, whereas ϑ(Gn) = α∗(Gn) = n2/2 by the standard argument of Cabello, Severini and Winter (CSW). Through the CSW correspondence, every member thus gives a state-independent noncontextuality inequality with classical bound (n2−1)/2 and quantum value n2/2. The list of orthogonal pairs, α, α∗ and the value of Lovász’s max-characterization of ϑ are formalized in Lean 4 with Mathlib, for the rays given by explicit coordinates.

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EXCEPTIONAL AND NON-CRYSTALLOGRAPHIC ROOT SYSTEMS AND THE KOCHEN-SPECKER THEOREM

Artur E. Ruuge

Abstract. The Kochen-Specker theorem states that a 3-dimensional complex Euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables (the orthogonal projectors) cannot be assigned with 0 and 1 values in a classically consistent way. This paper shows that the irreducible root systems of exceptional and of non-crystallographic types are useful in constructing such configurations in other dimensions. The cases E6 and E7 lead to new examples, while F4, E8, and H4, yield a new interpretation of the known ones. The described configurations have an additional property: they are saturated, i.e. the tuples of mutually orthogonal lines, being partially ordered by inclusion, yield a poset with all maximal elements having the same cardinality (the dimension of space). 1.

•Nov 27, 2012•5 citations
We develop a geometric framework for constructing and organizing Kochen--Specker configurations in real four-dimensional space. The construction uses finite grids on pairs of smooth quadrics related by Euclidean polarity. Their incidence geometry directly produces orthogonal measurement contexts and parity proofs of quantum contextuality, yielding infinite families of configurations and a geometric interpretation and extension of a previously known cyclic construction. For a distinguished subfamily, the same geometry admits a canonical completion determined by secant lines. The first two instances of this completion recover the exceptional root configurations of types F4F_4F4​ and H4H_4H4​, while the smallest case also recovers the Cabello configuration and its embedding in the Peres configuration. Thus several prominent four-dimensional contextual configurations, previously obtained from different constructions, arise from a single projective-geometric mechanism.
Logical Kochen-Specker (KS) contextuality is widely regarded as an intrinsic property of specially constructed measurement configurations. We show instead that it can emerge from KS-colorable vector sets through a constructive procedure we call the Hamilton extension. Defined for four-dimensional vector sets, the Hamilton extension associates each real vector with a measurement context while inducing additional measurement contexts among Hamilton-extended children of distinct parent vectors. These emergent contexts fundamentally alter the compatibility structure, transforming KS-colorable configurations into KS-uncolorable ones and recovering logical contextuality lost under apex-vertex augmentation. We establish a sharp and optimal threshold -- the Hamilton extension of every five-vector parent set remains KS-colorable, whereas suitably chosen six-vector parent sets already generate logical KS contradictions. Thus, six vectors constitute the smallest parent set capable of generating KS contradiction through this mechanism. Our results reveal a new structural route to contextuality, provide a systematic framework for constructing compact KS sets, and have implications for contextuality-based quantum information protocols and graph-theoretic approaches to nonclassicality.
Observable-based Kochen-Specker proofs are configurations of multi-qubit Pauli observables grouped into contexts whose products are plus or minus the indentity. Their robustness as state-independent contextuality tests can be measured by the tolerated error per context ε=2d/∣H∣\varepsilon = 2d/|H|ε=2d/∣H∣, where ddd is the contextuality degree and ∣H∣|H|∣H∣ the number of contexts. Since the degree depends only on an underlying abstract structure called the hypergram i.e. the pair formed by the context hypergraph and the anticommutation graph, the search for highly contextual proofs can be carried out on them instead, with no reference to qubits or to any particular Pauli labeling. We further exploit this by enumerating anticommutation graphs first, and then by associating to each graph GGG the single hypergram carrying its entire hypergraph support HS(G)HS(G)HS(G), so that exactly one candidate is examined per graph. Applied to the House of Graphs database and to censuses of vertex-transitive graphs on at most 24 vertices, this pipeline recovers the Peres-Mermin square, the doily and the Mermin pentagram, and yields configurations reaching ε=0.707\varepsilon = 0.707ε=0.707, against 0.4240.4240.424 for the previous published record. The best configurations are predominantly those stemming from line graphs and unions of graphs; we explain the former by showing that every perfect matching of a graph is a context of its line graph, which exhibits the Peres-Mermin square and the doily as the first members of two infinite families. We close with finite geometric descriptions of the most striking configurations inside symplectic polar spaces, in terms of ovoids, hyperbolic quadrics and Fano planes.
Contextuality is a feature differentiating between classical and quantum physics. It is anticipated that it may become an important resource for quantum computing and quantum information processing. Contextuality was asserted by the Kochen-Specker (KS) theorem. We study parity proofs of the KS theorem. Although many parity proofs exist, only finitely many of them have been discovered in any real or complex space of fixed dimension. We study a special family of chordal ring graphs. We construct orthonormal representations of their line graphs in four-dimensional real spaces. Our construction takes advantage of the high degree of symmetry present in the special class of chordal rings that we use. In this way we find, for the first time, an infinite family of KS sets in a fixed dimension.
A ``magic rectangle" of eleven observables of four qubits, employed by Harvey and Chryssanthacopoulos (2008) to prove the Bell-Kochen-Specker theorem in a 16-dimensional Hilbert space, is given a neat finite-geometrical reinterpretation in terms of the structure of the symplectic polar space W(7,2)W(7,2)W(7,2) of the real four-qubit Pauli group. Each of the four sets of observables of cardinality five represents an elliptic quadric in the three-dimensional projective space of order two (PG(3,2)(3,2)(3,2)) it spans, whereas the remaining set of cardinality four corresponds to an affine plane of order two. The four ambient PG(3,2)(3, 2)(3,2)s of the quadrics intersect pairwise in a line, the resulting six lines meeting in a point. Projecting the whole configuration from this distinguished point (observable) one gets another, complementary ``magic rectangle" of the same qualitative structure.
We study the orthogonality graphs Gn of the Kochen–Specker configurations Zn,n(kn, kn) recently obtained by Favacchio from grids on dual quadrics: for every odd n ≥ 3 they consist of 2n2 rays in R4 and n2 orthogonal bases, and G3 is the 18-ray set of Cabello, Estebaranz and García-Alcaine. We determine all orthogonal pairs, not only those inside the bases: Gn is (2n + 1)-regular, and the “only if” direction rests on the fact that 2 cos(2πt/n) is a unit in the ring of algebraic integers for odd n and t ̸≡ 0, while 2 is not. Consequently the independence number is α(Gn) = (n2 − 1)/2, with an explicit maximum independent set, whereas ϑ(Gn) = α∗(Gn) = n2/2 by the standard argument of Cabello, Severini and Winter (CSW). Through the CSW correspondence, every member thus gives a state-independent noncontextuality inequality with classical bound (n2−1)/2 and quantum value n2/2. The list of orthogonal pairs, α, α∗ and the value of Lovász’s max-characterization of ϑ are formalized in Lean 4 with Mathlib, for the rays given by explicit coordinates.
Abstract. The Kochen-Specker theorem states that a 3-dimensional complex Euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables (the orthogonal projectors) cannot be assigned with 0 and 1 values in a classically consistent way. This paper shows that the irreducible root systems of exceptional and of non-crystallographic types are useful in constructing such configurations in other dimensions. The cases E6 and E7 lead to new examples, while F4, E8, and H4, yield a new interpretation of the known ones. The described configurations have an additional property: they are saturated, i.e. the tuples of mutually orthogonal lines, being partially ordered by inclusion, yield a poset with all maximal elements having the same cardinality (the dimension of space). 1.