Exact Quantum Maxima of the $n$-Cycle Overlap Inequalities

Abstract

We derive the exact quantum maximum over all finite-dimensional quantum realizations of the $n$-cycle overlap inequalities, $S_n^{\max}=n\cos^2(π/(2n))-1$, valid for arbitrary cycle length $n\ge3$. The bound is saturated by an explicit family of coplanar qubit states equally spaced along a Fubini--Study geodesic of length $(n-1)π/(2n)$, establishing dimensional saturation of the overlap-cycle hierarchy. Thus, the global optimum over all finite-dimensional quantum realizations is already achieved in dimension two. For three paths, coherence-free models satisfy $\mathcal{V}_{12}^2+\mathcal{V}_{23}^2-\mathcal{V}_{13}^2\le1$, whereas quantum theory allows the larger value $5/4$. Within generalized noncontextuality frameworks, violations witness preparation contextuality. We further derive explicit visibility thresholds for arbitrary cycle length, identifying interference visibility as an operational probe of overlap-based nonclassicality, and discuss feasible photonic implementations.