2026/07/16 by Ravishankar Ramanathan
#quant-ph
Gleason's theorem identifies the Born rule via non-contextuality over an infinite continuous lattice of projectors, while its corollary the Kochen-Specker (KS) theorem rules out the specific class of deterministic (\0,1\-valued) noncontextual models using finite sets of projectors. Gleason's theorem also indicates the existence of KS-type finite vector constructions to rule out other discrete nondeterministic alternatives to quantum theory beyond the \0, 1\ case. Here, we construct finite measurement configurations to rule out noncontextual empirical models with outcome probabilities drawn from an arbitrary finite subset O ⊂ [0,1]. We do this by two means: (i) constructing a family of experimentally feasible state-dependent Hardy-type tests, and (ii) proving a generalized KS theorem for a broad class of O that includes prior results as special cases. In the sheaf-theoretic framework of Abramsky and Brandenburger, a hierarchy of probabilistic-possibilistic-strong contextuality has been established quantifying contextuality as a resource. We extend this framework by introducing strong O-valued contextuality, showing that quantum theory evades global sections of all finite-valued presheaves. We also discuss the implications of the result on finite many-valued logics as viable ontological models for quantum theory and for contextuality-based (semi)-device-independent protocols.