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arXiv:2509.08636

Construction of Kochen-Specker Sets from Mutually Unbiased Bases

Mirko Navara, Karl Svozil

Published Sep 10, 2025• 0 citations
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Abstract

We give a constructive analysis of Kochen-Specker (KS) contextuality at the level of complete orthogonal bases (contexts). In dimension three, three noncomputational mutually unbiased bases generate a union of 165 rays and 130 contexts. Exact provenance accounting gives a symmetric decomposition: the three 69-ray lineages share $21$ rays and $10$ contexts, while each contributes 48 exclusive rays and $40$ lineage-specific contexts. Each 69-ray, 50-context lineage is a KS configuration, and this framework clarifies the relation among the Yu-Oh, Harding-Salinas Schmeis, and Cabello constructions. In dimensions four and five, we give parametrized orthogonality gadgets whose connector contexts force a central ray to be maximally unbiased relative to the computational basis. For the concrete four-dimensional example, the 20-ray gadget and the Cabello 18-ray set are informationally equivalent subsets of the 24-ray Peres-Mermin eigensystem, although their context hypergraphs have different two-valued-state behavior. The results show why both the rays and the complete incidence structure of their contexts are needed in assessments of KS contextuality.

DOI
10.48550/arxiv.2509.08636

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We give a constructive analysis of Kochen-Specker (KS) contextuality at the level of complete orthogonal bases (contexts). In dimension three, three noncomputational mutually unbiased bases generate a union of 165 rays and 130 contexts. Exact provenance accounting gives a symmetric decomposition: the three 69-ray lineages share 212121 rays and 101010 contexts, while each contributes 48 exclusive rays and 404040 lineage-specific contexts. Each 69-ray, 50-context lineage is a KS configuration, and this framework clarifies the relation among the Yu-Oh, Harding-Salinas Schmeis, and Cabello constructions. In dimensions four and five, we give parametrized orthogonality gadgets whose connector contexts force a central ray to be maximally unbiased relative to the computational basis. For the concrete four-dimensional example, the 20-ray gadget and the Cabello 18-ray set are informationally equivalent subsets of the 24-ray Peres-Mermin eigensystem, although their context hypergraphs have different two-valued-state behavior. The results show why both the rays and the complete incidence structure of their contexts are needed in assessments of KS contextuality.
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.
Quantum contextuality supports quantum computation and communication. One of its main vehicles is hypergraphs. The most elaborated are the Kochen-Specker ones, but there is also another class of contextual sets that are not of this kind. Their representation has been mostly operator-based and limited to special constructs in three- to six-dim spaces, a notable example of which is the Yu-Oh set. Previously, we showed that hypergraphs underlie all of them, and in this paper, we give general methods-whose complexity does not scale up with the dimension-for generating such non-Kochen-Specker hypergraphs in any dimension and give examples in up to 16-dim spaces. Our automated generation is probabilistic and random, but the statistics of accumulated data enable one to filter out sets with the required size and structure.
Kochen-Specker (KS) theorem reveals the inconsistency between quantum theory and any putative underlying model of it satisfying the constraint of KS-noncontextuality. A logical proof of the KS theorem is one that relies only on the compatibility relations amongst a set of projectors (a KS set) to witness this inconsistency. These compatibility relations can be represented by a hypergraph, referred to as a contextuality scenario. Here we consider contextuality scenarios that we term KS-uncolourable, e.g., those which appear in logical proofs of the KS theorem. We introduce a hypergraph framework to obtain noise-robust witnesses of contextuality from such scenarios. Our approach builds on the results of R. Kunjwal and R. W. Spekkens, Phys. Rev. Lett. 115, 110403 (2015), by providing new insights into the relationship between the structure of a contextuality scenario and the associated noise-robust noncontextuality inequalities that witness contextuality. The present work also forms a necessary counterpart to the framework presented in R. Kunjwal, Quantum 3, 184 (2019), which only applies to KS-colourable contextuality scenarios, i.e., those which do not admit logical proofs of the KS theorem but do admit statistical proofs. We rely on a single hypergraph invariant, defined in R. Kunjwal, Quantum 3, 184 (2019), that appears in our contextuality witnesses, namely, the weighted max-predictability. The present work can also be viewed as a study of this invariant. Significantly, unlike the case of R. Kunjwal, Quantum 3, 184 (2019), none of the graph invariants from the graph-theoretic framework for KS-contextuality due to Cabello, Severini, and Winter (the ``CSW framework", Phys. Rev. Lett. 112, 040401 (2014)) are relevant for our noise-robust noncontextuality inequalities.
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.
A Kochen-Specker (KS) set is a specific set of projectors and measurement contexts that prove the Bell-Kochen-Specker contextuality theorem. The simplest known KS sets in Hilbert space dimensions d=3,4,5,6,8d=3,4,5,6,8d=3,4,5,6,8 are reproduced, and several methods by which a new KS set can be constructed using one or more known KS sets in lower dimensions are reviewed and improved. These KS sets and improved methods enable the construction of explicitly critical new KS sets in all dimensions, where critical refers to the irreducibility of the set of contexts. The simplest known critical KS sets are derived in all even dimensions $d\ensuremath{\ge}10$ with at most nine contexts and 30 projectors, and in all odd dimensions $d\ensuremath{\ge}7$ with at most 13 contexts and 39 projectors. These results show that neither the number of contexts nor the number of projectors in a minimal KS set scales with dimension ddd.
Quantum contextuality turns out to be a necessary resource for universal quantum computation and important in the field of quantum information processing. It is therefore of interest both for theoretical considerations and for experimental implementation to find new types and instances of contextual sets and develop methods of their optimal generation. We present an arbitrarily exhaustive hypergraph-based generation of the most explored contextual sets [Kochen-Specker (KS) ones] in 4, 6, 8, 16, and 32 dimensions. We consider and analyze 12 KS classes and obtain numerous properties of theirs, which we then compare with the results previously obtained in the literature. We generate several thousand additional types and instances of KS sets, including all KS sets in three of the classes and the upper part of a fourth set. We make use of the McKay-Megill-Pavi\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \acute{c}\else \'{c}\fi{} (MMP) hypergraph language, algorithms, and programs to generate KS sets strictly following their definition from the Kochen-Specker theorem. This approach proves to be particularly advantageous over the parity-proof-based ones (which prevail in the literature) since it turns out that only a very few KS sets have a parity proof (in six KS classes 0.01% and in one of them 0%). MMP hypergraph formalism enables a translation of an exponentially complex task of solving systems of nonlinear equations, describing KS vector orthogonalities, into a statistically linearly complex task of evaluating vertex states of hypergraph edges, thus exponentially speeding up the generation of KS sets and enabling us to generate billions of novel instances of them. The MMP hypergraph notation also enables us to graphically represent KS sets and to visually discern their features.