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LIT-266: Contextuality for preparations, transformations, and unsharp measurements

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Properties5
StatusActive — read 2026-10-14 (NOTE-748) in full; worth reading as the source of generalized noncontextuality and as the bridge between it and the Kochen–Specker notion. It proves preparation, transformation and unsharp-measurement contextuality for a qubit, shows the traditional notion is measurement noncontextuality plus outcome determinism for sharp measurements, and proves that preparation noncontextuality implies that determinism. Read it knowing that every result assumes exact operational equivalence and gives no inequality; that the bridge (§VIII.A) uses the full set of quantum preparations; and that "any classical theory is necessarily noncontextual" is a footnote assertion. It argues from ontological economy and does not mention Leibniz.
Version2
Tagscontextuality, quantum-foundations
Cited as “paper” byNOTE-238 — Spekkens, generalized contextuality (2005), NOTE-748 — Spekkens — Contextuality for preparations, transformations, and unsharp measurements
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-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-231 — Within classical realism, a generalized-noncontextuality no-go refutes Leibnizian representations, not realist representations as such; it rests on a stronger assumption than a Bell-like no-go and so applies where none exists, as in prepare-measure scenarios, but only Leibnizianity implying noncontextuality is proved

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R. W. Spekkens (2004), Physical Review A 71, 052108 (2005) — arXiv:quant-ph/0406166

Key takeaways

From the reading, NOTE-748.

  • The definition. “A noncontextual ontological model of an operational theory is one wherein if two experimental procedures are operationally equivalent, then they have equivalent representations in the ontological model.” Equivalence quantifies over every procedure of the other kinds (Eqs. 1–3). The context is whatever the equivalence class leaves unspecified.
  • The old notion inside the new. For sharp measurements with idempotent indicator functions, independence from the fine-graining of the PVM “is just the traditional notion of noncontextuality” (§III). Fine-grainings of non-trivial PVMs do not exist in dimension two, so the traditional proofs need dimension three.
  • A qubit is preparation contextual (§IV). Six pure states give five decompositions of I/2. Orthogonal states have disjoint supports, and preparation noncontextuality then forces every distribution to vanish. No measurement assumption is used.
  • And transformation contextual (§VI), by the same structure on six rotations about one axis. It is also contextual for unsharp measurements if sharp measurements are outcome-deterministic (§V).
  • The bridge (§VIII.A). Preparation noncontextuality implies outcome determinism for every sharp measurement. The proof uses every density operator’s membership in some decomposition of I/d. So Kochen–Specker proofs refute universal noncontextuality.
  • Measurement noncontextuality alone is consistent with quantum theory (§VIII.B). The Beltrametti–Bugajski model shows it: “one cannot confine all the contextuality into the measurements” (§IX).
  • Limits it states itself. Exact equivalence may never hold under finite precision (§IX). That classical theories are always noncontextual is asserted in a footnote.

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), and read in full on 2026-10-14 in NOTE-748. It is the source of THEORY-011 and THEORY-015 and bears on QUESTION-067. It is the only held work that states the link between Kochen–Specker contextuality and the generalized notion that LIT-003, LIT-007 and LIT-019 rely on. The further join to the sheaf form (LIT-016) is THEORY-015’s own. The reading confirms both theories’ accounts of it. It also answers the three checks the seeding set: §VIII.A needs the full preparation set, §V’s proofs assume outcome determinism for sharp measurements, and the classical-implies-noncontextual footnote is asserted. It corrects one detail: the qubit has no Kochen–Specker proof only for sharp measurements, since Cabello’s POVM proof exists and is rejected here for its assumption, not its mathematics. Its process-theoretic successor is LIT-945.

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