Chromatic quantum contextuality is a criterion of quantum nonclassicality based on (hyper)graph coloring constraints. If a quantum hypergraph requires more colors than the number of outcomes per maximal observable (context), it lacks a classical realization with n-uniform outcomes per context. Consequently, it cannot represent a “completable” noncontextual set of coexisting n-ary outcomes per maximal observable. This result serves as a chromatic analogue of the Kochen-Specker theorem. We present an explicit example of a four-colorable quantum logic in dimension three. Furthermore, chromatic contextuality suggests a novel restriction on classical truth values, thereby excluding two-valued measures that cannot be extended to n-ary colorings. Using this framework, we establish new bounds for the house, pentagon, and pentagram hypergraphs, refining previous constraints.
Intertwined complete projective experiments share outcomes, motivating the search for a common spectral labeling across contexts. Strong coloring tests whether complete contexts admit one context-independent nondegenerate spectrum; when that labeling exists, chromatic separability tests which event distinctions it preserves. We distinguish these questions from ordinary Kochen--Specker statelessness, geometric realizability, event-order representation, and Born-probability models. A state extends to a strong d-coloring exactly when deletion of its support leaves a (d-1)-colorable graph. Faithful real 17- and 21-ray restrictions of completed Yu--Oh have separating ordinary states but forced color identities; one quantum preparation requires exactly 1/9 nonextendable weight despite admitting an ordinary-state model. Independent Hermitian projection arguments correct earlier inferences from a chromatic gap to statelessness or nonrealizability. Hesse's complex 37-ray example separates real and complex geometry and has exact state-family minima 6/7/18 for three representation requirements. A rational two-plane construction gives a linear real 65-ray example with 393,216 states and explicit event-order and probability failures. General and specified ancestry results distinguish geometric KS extensions from abstract chromatic-to-stateless constructions. Detailed enumerations, regenerated state certificates, and extended proofs are supplementary.
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.
Cameron, Montanaro, Newman, Severini and Winter gave a construction which shows that, for
n∈{2,4,8}, any graph
G which admits a real
n-dimensional orthogonal representation is quantumly
n-colorable. This result can be recast as the statement that the real sphere
Sn−1 is quantumly
n-colorable for these values of
n. We investigate possible extensions of their construction. We first show that their hypothesis that the orthogonal representation be real-valued is required by proving that there is no analogue of this for the complex spheres, which all have quantum chromatic number strictly bigger than the dimension except in two dimensions. We also provide candidate finitary witnesses of this and show for the first time that the real and complex orthogonal ranks are distinct as a byproduct. For the real case, we show that if
Sn−1 is quantumly
n-colorable, then either
n=2 or
n is a multiple of 4, and show that the converse holds whenever a Hadamard matrix of order
n exists. Hence, assuming the Hadamard conjecture, this completely classifies the dimensions to which the CMNSW construction can be extended. Our method of proof involves showing the equivalence between the existence of such a construction and the existence of a maximal code space for Clifford-algebraic errors given a clean ancilla, and we believe that the representation-theoretic techniques we use for tackling the latter problem could be of independent interest. It also follows from this equivalence that
Sn−1 admits a rank-one quantum
n-coloring if and only if
n∈{2,4,8}, thereby settling a conjecture of Zeng and Zhang, as does the fact that for all
m≥1, there exists a catalytic zero-error remote state preparation protocol for real
m-qubit states with
m bits of communication and which consumes
m ebits.
We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph.After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is
2, nor if it is
3 in a restricted quantum model; on the other hand, we exhibit a graph on
18 vertices and
44 edges with chromatic number
5 and quantum chromatic number
4.
Hypergraph states are multi-qubit states that form a subset of the locally maximally entangleable states and a generalization of the well--established notion of graph states. Mathematically, they can conveniently be described by a hypergraph that indicates a possible generation procedure of these states; alternatively, they can also be phrased in terms of a non-local stabilizer formalism. In this paper, we explore the entanglement properties and nonclassical features of hypergraph states. First, we identify the equivalence classes under local unitary transformations for up to four qubits, as well as important classes of five- and six-qubit states, and determine various entanglement properties of these classes. Second, we present general conditions under which the local unitary equivalence of hypergraph states can simply be decided by considering a finite set of transformations with a clear graph-theoretical interpretation. Finally, we consider the question whether hypergraph states and their correlations can be used to reveal contradictions with classical hidden variable theories. We demonstrate that various noncontextuality inequalities and Bell inequalities can be derived for hypergraph states.