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.
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.
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.
Contextuality is a key distinguishing feature between classical and quantum physics. It expresses a fundamental obstruction to describing quantum theory using classical concepts. In turn, when understood as a resource for quantum computation, it is expected to hold the key to quantum advantage. Yet, despite its long recognised importance in quantum foundations and, more recently, in quantum computation, the mathematics of contextuality has remained somewhat elusive - different frameworks address different aspects of the phenomenon, yet their precise relationship often is unclear. In fact, there is a glaring discrepancy already between the original notion of contextuality introduced by Kochen and Specker on the one side [J. Math. Mech., 17, 59, (1967)], and the modern approach of studying contextual correlations on the other [Rev. Mod. Phys., 94, 045007 (2022)]. In a companion paper [arXiv:2408.16764], we introduce the conceptually new tool called ``context connections'', which allows to cast and analyse Kochen-Specker (KS) contextuality in new form. Here, we generalise this notion, and based on it prove a complete characterisation of KS contextuality for finite-dimensional systems. To this end, we develop the framework of ``observable algebras". We show in detail how this framework subsumes the marginal and graph-theoretic approaches to contextuality, and thus that it offers a unified perspective on KS contextuality. In particular, we establish the precise relationships between the various notions of ``contextuality" used in the respective settings, and in doing so, generalise a number of results on the characterisation of the respective notions in the literature.
We look at generalisations of sets of vectors proving the Kochen-Specker theorem in 3 and 4 dimensions. It has been shown that two such sets, although unitarily inequivalent, are part of a larger 3-parameter family of vectors that share the same orthogonality graph. We find that these sets are unusual, in that the vectors in all other Kochen-Specker sets investigated here are fully determined by orthog-onality conditions and thus admit no free parameters. The Kochen-Specker (KS) theorem [1] is a statement about the fundamen-tal nature of quantum mechanics: no non-contextual hidden variable theory can reliably reproduce quantum mechanical predictions. The KS theorem states that in a Hilbert space with dimension n ≥ 3 it is impossible to assign definite values v ∈ {0, 1} to m projection operators Pm so that for every commuting set of projection operators satisfying Pm = I, the correspond-ing definite values obey v (Pm) = 1. The KS theorem is often visualised by means of a set of "uncolourable" vectors, where projection operators are represented by their constituent rays in Hilbert space and rays can be simplified to vectors as only directions are relevant here. The vectors are coloured according to their pre-assigned val-ues and the summation conditions above translate into colouring rules. This allows us to construct orthogonality graphs that, when uncolourable, pro-vide a contradiction within non-contextual hidden variable models and thus a proof of the KS theorem. Sets of vectors exemplifying the KS theorem have been discovered by nu-merous authors, with an aim to reduce the number of vectors required to reach a contradiction. Most notably, in a 3-dimensional Hilbert space, the original KS proof of 117 vectors has been replicated with sets using between 31 and 33 vectors [2, 3, 4, 5] and in a 4-dimensional Hilbert space, sets of between 24 and the lower limit of 18 vectors have been realised [2, 6, 7]. Additionally, an extensive computer search has recently revealed over one thousand of such KS sets in R4 [8]. In a publication by E. Gould and P. K. Aravind [10], two of these sets in 3 dimensions have been shown to share the same orthogonality graph. Specifically, the sets of 33 vectors found by Penrose [3] and Peres [2] have
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.
For the first time we construct an infinite family of Kochen-Specker sets in a space of fixed dimension, namely in R4. While most of the previous constructions of Kochen-Specker sets have been based on computer search, our construction is analytical and it comes with a short, computer-free proof.
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 F4 and H4, 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.