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LIT-263: Kochen-Specker contextuality

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Properties5
StatusActive — read 2026-10-14 (NOTE-844) in full; worth reading as the record's reference for Kochen–Specker contextuality: the marginal-problem definition (Fine's theorem) with the sheaf form among its equivalents, why contexts need more than joint measurability, how experiments model disturbance, and Vorob'ev's acyclicity theorem. Read it knowing that it is a review whose results are mostly cited, that it describes Spekkens's notion only to set it apart and never derives the relation between the two, and that its account of maximally noncontextual models is one noise model among several, with no behavioural data.
Version2
Tagscontextuality, quantum-foundations
Cited as “paper” byNOTE-237 — Budroni et al., Kochen–Specker contextuality (RMP review), NOTE-844 — Budroni, Cabello, Gühne, Kleinmann & Larsson — Kochen-Specker Contextuality
Cited as “source” byTHEORY-011 — Generalized contextuality is strictly broader than Kochen–Specker contextuality: a qubit already has no preparation-noncontextual model, though no Kochen–Specker argument with sharp measurements exists in dimension two, THEORY-013 — Once context-dependent marginals are separated from contextuality, the behavioural, social and word-meaning data published as contextual show none, THEORY-015 — Kochen–Specker noncontextuality is measurement noncontextuality plus outcome determinism for sharp measurements, and preparation noncontextuality implies that determinism, so every Kochen–Specker proof refutes universal generalized noncontextuality while measurement noncontextuality alone is consistent with quantum theory, THEORY-303 — A measurement scenario whose compatibility hypergraph is acyclic admits no contextuality: locally consistent context distributions always glue to a global one, so a contextual scenario needs an induced cycle of length at least four, THEORY-309 — If contexts are defined by joint measurability alone, every nondisturbing correlation, maximally contextual ones included, is realized by trivial measurements on a one-dimensional quantum system, so Kochen–Specker contextuality needs a stronger operational notion of context

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Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, Jan-Åke Larsson (2021), Reviews of Modern Physics 94, 045007 (2022) — ARXIV-2102.13036

Key takeaways

From the reading, NOTE-844.

  • Definition (§IV.A.1). An NCHV model (Eq. 15) exists iff a global distribution reproduces the context marginals (Eq. 16, Fine; Suppes & Zanotti). Local consistency is “sometimes called the sheaf condition”. The marginal-problem, polytope, sheaf, graph and hypergraph formulations are asserted to be substantially equivalent.
  • Contexts (§IV.B.2). Adapting Fritz (2012), POVMs p(ab|xy)𝟙 on a one-dimensional space realize every nonsignalling, and in general every nondisturbing, behaviour as jointly measurable. So joint measurability alone trivializes contextuality. The review defines contexts by sequences of outcome-repeatable, nondisturbing measurements.
  • Imperfections (§IV.C). Four noise models, each an explicit assumption: cumulative noise, context-independent evolution, ε-faithfulness, and Kujala et al.’s maximally noncontextual models. The KCBS data stay contextual under the last (Eq. 55).
  • Structure (§V.B.2). Vorob’ev: consistent marginals on an acyclic hypergraph always glue. Contextuality needs an induced cycle of length ≥ 4, so CHSH and KCBS are the smallest contextual scenarios.
  • Spekkens (§IV.E). The qubit’s preparation contextuality is restated (Eqs. 67–70). Mazurek et al.’s inequality uses preparation and measurement noncontextuality together, and its experimental violation assumes tomographic completeness. KS’s and Bell’s qubit models are preparation contextual.
  • Applications (§VI). For odd prime d, stabilizer-measurement contextuality coincides with Wigner negativity (Howard et al.). The full stabilizer subtheory has an efficient contextual simulation (Hindlycke & Larsson).

Standing in the record

Filed on 2026-09-27 while bringing the record’s contextuality threads together at the owner’s request (see the curation entry of that date). Read on 2026-10-14 in NOTE-844, which supersedes the seeded skim NOTE-237.

It is cited by THEORY-011, THEORY-012, THEORY-013 and THEORY-015, and the reading checks each citation. THEORY-012 and THEORY-013 are confirmed, with precisions. THEORY-011 leans on an experiment that tests preparation and measurement noncontextuality jointly. THEORY-015’s support is thin: the review does not state its bridge. Two accounts were filed from the reading. THEORY-309: joint measurability alone trivializes contextuality. THEORY-303: acyclic scenarios cannot be contextual.

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