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arXiv:2609.20205

Complex coordinates forced by finite orthogonality hypergraphs in dimension three

Mirko Navara, Karl Svozil

math.COquant-phSubmitted Jul 25, 2026• Announced Jul 28, 2026• 0 citations
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Abstract

We exhibit two finite orthogonality hypergraphs that admit faithful orthogonal representations by rays in complex three-dimensional space but admit none in real three-dimensional space. Thus, although each orthogonality diagram is unchanged, its coordinatizability in dimension three depends on whether the scalar field is complex or real. The first hypergraph has 20 vertices and 12 contexts. The second hypergraph adds one vertex and two contexts. Phase-adjusted realification also gives explicit faithful orthogonal representations of both hypergraphs in real six-dimensional space.

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8 pages, 2 figures

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We exhibit two finite orthogonality hypergraphs that admit faithful orthogonal representations by rays in complex three-dimensional space but admit none in real three-dimensional space. Thus, although each orthogonality diagram is unchanged, its coordinatizability in dimension three depends on whether the scalar field is complex or real. The first hypergraph has 20 vertices and 12 contexts. The second hypergraph adds one vertex and two contexts. Phase-adjusted realification also gives explicit faithful orthogonal representations of both hypergraphs in real six-dimensional space.
We describe a phase-adjusted realification procedure that embeds any finite set of rays in C^3 into R^6. By assigning an appropriate phase to each ray before applying the standard coordinate-wise map, we can arrange that two rays are orthogonal in C^3 if and only if their images are orthogonal in R^6, so the construction yields a faithful orthogonal representation of the original complex configuration. As a concrete example, we consider the 165 projectively distinct rays used in a C^3 Kochen-Specker configuration obtained from mutually unbiased bases, list these 165 rays explicitly in C^3, and give for each of them its image in R^6 under the canonical realification map. We also note that, because the original 3-element contexts are no longer maximal in R^6, the embedded configuration admits two-valued states even though its realisation with maximal contexts in C^3 is Kochen-Specker uncolourable.
We discuss representations and colorings of orthogonality hypergraphs in terms of their two-valued states interpretable as classical truth assignments. Such hypergraphs, if they allow for a faithful orthogonal representation, have quantum mechanical realizations in terms of intertwined contexts or maximal observables that are widely discussed as empirically testable criteria for contextuality. A reconstruction is possible for the class of perfectly separable hypergraphs. Colorings can be constructed from a minimal set of two-valued states. Some examples from exempt categories are presented, which either cannot be reconstructed by two-valued states or whose two-valued states cannot yield a chromatic number that is equal to the maximal clique number.
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.
In this paper, we study linear operators on real and complex Euclidean spaces which are real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We note the difference between properties of real-orthogonal projections on real and on complex spaces. We can compare some partial order properties of orthogonal and of real-orthogonal projections. We prove that the set of all real-orthogonal projections in a finite-dimensional complex or real space is a quantum pseudo-logic.
This paper presents the quantum Möbius-Escher-Penrose hypergraph, drawing inspiration from paradoxical constructs such as the Möbius strip and Penrose's “impossible objects.” The hypergraph is constructed using faithful orthogonal representations in Hilbert space, thereby embedding the graph within a quantum framework. Additionally, a quasiclassical realization is achieved through two-valued states and partition logic, leading to an embedding within a Boolean algebra. This dual representation delineates the distinctions between classical and quantum embeddings, with a particular focus on contextuality, highlighted by violations of exclusivity and completeness, quantified through classical and quantum probabilities. The paper also examines violations of Boole's conditions of possible experience using correlation polytopes, underscoring the inherent contextuality of the hypergraph. These results offer deeper insights into quantum contextuality and its intricate relationship with classical logic structures.
Quantum contextuality plays a significant role in supporting quantum computation and quantum information theory. The key tools for this are the Kochen-Specker and non-Kochen-Specker contextual sets. Traditionally, their representation has been predominantly operator-based, mainly focusing on specific constructs in dimensions ranging from three to eight. However, nearly all of these constructs can be represented as low-dimensional hypergraphs. This study demonstrates how to generate contextual hypergraphs in any dimension using various methods, particularly those that do not scale in complexity with increasing dimensions. Furthermore, we introduce innovative examples of hypergraphs extending to dimension 32. Our methodology reveals the intricate structural properties of hypergraphs, enabling precise quantifications of contextuality. Additionally, we investigate several promising applications of hypergraphs in quantum communication and quantum computation, paving the way for future breakthroughs in the field.