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THEORY-016: An operational theory admits a generalized-noncontextual model exactly when its GPT admits a positive quasiprobability representation, and for a tomographically local theory any diagram-preserving such model is an exact frame with exactly as many ontic states as the GPT's dimension

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Properties6
StatusActive
Version3
Tagscontextuality, quantum-foundations, mathematics
SourceLIT-003 — A structure theorem for generalized-noncontextual ontological models, LIT-007 — Quasi-quantization: classical statistical theories with an epistemic restriction, LIT-073 — General probabilistic theories: An introduction
Extended byTHEORY-014 — In both formalisms of contextuality, classicality is the existence of a nonnegative version of a linear representation that always exists over the reals, so negativity is universal and only its unavoidability is nonclassical, 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
AnswersQUESTION-246 — In quantum theory, what does a failure of Kochen–Specker noncontextuality refute, as against a failure of Spekkens's generalized noncontextuality, and which of the two notions is broader?

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Source

Schmid et al. (2020), LIT-003, Def. 3.1, Props 3.2 and 3.4, Cor 3.5, Thm 2.8, Thm 4.1 (proof in App. B), Cor 4.2, Prop 4.3, §5.2, App. A (NOTE-017). Worked instance: Spekkens (2014), LIT-007, §IV.A. Background on GPTs: LIT-073. Schmid, Selby and Spekkens, LIT-945, restate the equivalence, citing LIT-003; it is a restatement, not an independent proof (NOTE-737).

What was actually shown

Spekkens’s generalized noncontextuality asks that operationally equivalent procedures — ones no preparation or measurement can tell apart — receive identical representations in the ontological model. LIT-003 states it for arbitrary compositional scenarios (Def. 3.1) and proves that a noncontextual ontological model of an operational theory exists iff its GPT admits an ontological model (a simplex embedding) iff it admits a positive quasiprobabilistic model (Props 3.2, 3.4, Cor 3.5), with no tomographic-locality assumption. For tomographically local GPTs, every diagram-preserving quasiprobabilistic map has the form M(T) = χ_B ∘ T ∘ χ_A⁻¹ (Thm 4.1), so a noncontextual model has |Λ_A| = dim A (Cor 4.2) and is an exact, non-overcomplete frame (Prop 4.3); §5.2 shows tomographic locality is needed for the count. A GPT has no contexts at all (App. A), which is where Kochen–Specker measurement contexts become invisible.

Spekkens’s quadrature/stabilizer subtheory for odd d is a worked instance: its Wigner representation is an ontological model (LIT-007 §IV.A), hence noncontextual, and NOTE-017 checked the d^(2n) count for small cases.

What this does not say

  • How to decide whether a positive representation exists (§4.5).
  • Anything for overcomplete frames (the Q and P functions) or excess-baggage models, which diagram preservation rules out by fiat, or for theories without tomographic locality, such as real quantum theory.
  • That the ontic states are unique up to relabelling; uniqueness is up to an invertible real linear map (Thm 4.7).
  • That Catani et al.’s toy field theory (LIT-019) is noncontextual; that paper asserts it without proof (NOTE-018).

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