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arXiv:2605.14395

Exact Quantum Maxima of the nnn-Cycle Overlap Inequalities

Mohd Asad Siddiqui

quant-phSubmitted May 14, 2026• Announced May 15, 2026• 0 citations• Revised Jun 9, 2026 (v3)
v1 · May 14, 2026v2 · May 15, 2026v3 · Jun 9, 2026v4 · Sep 30, 2026
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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.

DOI
10.1098/rspa.2026.0596
Comments
9 pages, 1 figure, Substantially revised version; exact quantum maximum established

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The problem of separating classical from quantum correlations is, in general, intractable and has been solved explicitly only in a few cases. In particular, known methods cannot provide general solutions for an arbitrary number of settings. We provide the complete characterization of the classical correlations and the corresponding maximal quantum violations for the case of $n\ensuremath{\ge}4$ observables ${X}_{0},...,{X}_{n\ensuremath{-}1}$, where each consecutive pair Xi,Xi+1{{X}_{i},{X}_{i+1}}Xi​,Xi+1​, sum modn\mathrm{mod}\phantom{\rule{0.16em}{0ex}}nmodn, is jointly measurable. This generalizes both the Clauser-Horne-Shimony-Holt and the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky scenarios, which are the simplest ones for locality and noncontextuality, respectively. In addition, we provide explicit quantum states and settings with maximal quantum violation and minimal quantum dimension.
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