The Quad-$C_5$ Graph: Maximum Contextuality Gap on Eight Vertices

Abstract

Quantum contextuality rules out classical descriptions in which measurement outcomes are independent of the compatible measurements performed alongside them. While the KCBS pentagon provides the minimal qutrit test, identifying larger but still compact contextuality witnesses remains important for understanding how contextuality can be organized, amplified, and benchmarked. We solve the eight-event version of this problem within the graph-theoretic framework for contextuality. An exhaustive search over all 11,117 connected non-isomorphic graphs with eight vertices identifies a sparse ten-edge graph, which we call Quad-$C_5$, as the maximum-gap witness. The graph has a larger absolute contextuality gap than the Wagner graph, a standard eight-vertex benchmark, despite having fewer exclusivity relations. Its structure is transparent: it is formed from four overlapping KCBS pentagons, with each edge shared by two pentagons. We prove that Quad-$C_5$ is already contextual for a single qutrit by giving an explicit three-dimensional construction. In this qutrit realization its contextuality margin equals that of KCBS, whereas its full quantum advantage is reached numerically in four dimensions and exceeds that of the Wagner graph. Quad-$C_5$ therefore provides a compact bridge between minimal qutrit contextuality and stronger four-dimensional contextuality witnesses.