The Quad-C5 Graph: Maximum Contextuality Gap on Eight Vertices
Abstract
Quantum measurements can exhibit contextuality: their outcomes cannot always be explained by assigning pre-existing values that are independent of which compatible measurements are performed together. The Klyachko-Can-Binicioğlu-Shumovsky (KCBS) inequality provides the canonical minimal test of this effect for a single three-level quantum system, or qutrit, using five measurement events arranged as a pentagon. Here we ask whether a larger but still compact set of measurement events can produce a stronger separation between quantum predictions and the corresponding noncontextual limit. We perform an exhaustive search over all 11,117 connected non-isomorphic graphs with eight vertices, where each vertex represents a measurement event and edges connect pairs of events that cannot occur together. We identify a sparse ten-edge graph, which we call Quad-C5, as the unique maximizer of this separation at the reported numerical precision. The graph can be understood as four overlapping KCBS pentagons, with every edge shared by two pentagons. Quad-C5 already demonstrates contextuality in a qutrit, for which we obtain an exact analytical result and find the same violation above the noncontextual bound as in the original KCBS test. The larger quantum-noncontextual separation allowed by the graph, however, becomes accessible in a four-level quantum system, where numerical optimization reaches the full graph-theoretic quantum bound and yields a larger contextuality gap than the standard eight-vertex Wagner-graph benchmark while requiring fewer pairwise constraints. Quad-C5 therefore provides a compact connection between minimal qutrit contextuality and stronger contextuality tests available in higher-dimensional quantum systems.
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BibTeX
@misc{tamer2026quadc5,
title = {{The Quad-$C_5$ Graph: Maximum Contextuality Gap on Eight Vertices}},
author = {Uğur Tamer and Berke Ünal and Alper Dizdar and Zafer Gedik and Özgür E. Müstecaplıoğlu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.12828},
eprint = {2605.12828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.12828},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Quad-C_5 Graph: Maximum Contextuality Gap on Eight Vertices
AU - Uğur Tamer
AU - Berke Ünal
AU - Alper Dizdar
AU - Zafer Gedik
AU - Özgür E. Müstecaplıoğlu
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2605.12828
UR - https://arxiv.org/abs/2605.12828
AB - Quantum measurements can exhibit contextuality: their outcomes cannot always be explained by assigning pre-existing values that are independent of which compatible measurements are performed together. The Klyachko-Can-Binicioğlu-Shumovsky (KCBS) inequality provides the canonical minimal test of this effect for a single three-level quantum system, or qutrit, using five measurement events arranged as a pentagon. Here we ask whether a larger but still compact set of measurement events can produce a stronger separation between quantum predictions and the corresponding noncontextual limit. We perform an exhaustive search over all 11,117 connected non-isomorphic graphs with eight vertices, where each vertex represents a measurement event and edges connect pairs of events that cannot occur together. We identify a sparse ten-edge graph, which we call Quad-C_5, as the unique maximizer of this separation at the reported numerical precision. The graph can be understood as four overlapping KCBS pentagons, with every edge shared by two pentagons. Quad-C_5 already demonstrates contextuality in a qutrit, for which we obtain an exact analytical result and find the same violation above the noncontextual bound as in the original KCBS test. The larger quantum-noncontextual separation allowed by the graph, however, becomes accessible in a four-level quantum system, where numerical optimization reaches the full graph-theoretic quantum bound and yields a larger contextuality gap than the standard eight-vertex Wagner-graph benchmark while requiring fewer pairwise constraints. Quad-C_5 therefore provides a compact connection between minimal qutrit contextuality and stronger contextuality tests available in higher-dimensional quantum systems.
ER -