THEORY-315: For a state whose four-state reflection orbit is realizable, generalized noncontextuality bounds two binary predictabilities by |⟨X⟩| + |⟨Z⟩| ≤ 1, so the form of a predictability trade-off, not its existence, can witness contextuality

Source

Catani, Leifer, Scala, Schmid and Spekkens (2022), Physical Review Letters 129, 240401, LIT-980, read in full in NOTE-825: Eqs. 1–13, Appendix A (the proof), Appendix B (the linear-program cross-check), Appendix C (three measurements).

What was actually shown

The setting. A prepare-measure scenario with two binary measurements X and Z and four preparations. Noncontextuality is Spekkens’s, for preparations only: operationally equivalent preparations, those with the same statistics for every measurement, get the same distribution over ontic states (Eq. 8). A state s1 is A²₁-orbit-realizable when there are states s2, s3, s4 with the same |⟨X⟩| and |⟨Z⟩| and the other three sign patterns (Eq. 10), and ½s1 + ½s3 = ½s2 + ½s4 (Eq. 11).

The inequality (Eq. 12, Appendix A). For such a state, every preparation-noncontextual model satisfies |⟨X⟩| + |⟨Z⟩| ≤ 1. The proof is analytic and short. With four outcome-deterministic ontic states and μ1 = (a, b, c, d), the constraints give ⟨X⟩ + ⟨Z⟩ = 2(b − c) ≤ 1 − c ≤ 1 (Eq. A33). The bound is tight (Eqs. A35–A38). It is also the reduction, under the orbit condition, of the eight inequalities Schmid, Spekkens and Wolfe derived by linear program for the same equivalence (Appendix B). This is what could have come out otherwise: a noncontextual model exceeding the diamond would refute it, and the explicit parameterization rules one out.

From one state to a theory. In a theory where every state is orbit-realizable (A²₁ symmetry), the inequality bounds the uncertainty relation itself. The qubit has the symmetry and the relation ⟨X⟩² + ⟨Z⟩² ≤ 1, a circle, so it violates the bound everywhere except where one predictability is 1, by up to √2. The qubit stabilizer theory saturates the bound, and it has a noncontextual model. The gbit, also symmetric, reaches 2. The simplicial theory of two classical bits has the gbit’s relation but not the symmetry. The states that reach joint predictability 1, its vertices, have equal-predictability counterparts but fail Eq. 11, so the bound does not apply to them.

So: an uncertainty relation as such does not mark nonclassicality, since the stabilizer theory has one and a noncontextual model. What marks it, given the symmetry, is joint predictability above the linear curve.

What this does not say

• It does not say that any trade-off curve above the line refutes noncontextuality. The bound holds only for orbit-realizable states. Without the equivalence of Eq. 11, perfect joint predictability is classical, as the simplicial theory shows.
• It does not say that unpredictability is classical and predictability nonclassical in general. The authors’ reversal holds within theories with A²₁ symmetry.
• It does not treat inexact equivalence. Eq. 11 is used as an equality. The letter says the inequality “is noise-robust, and can therefore be tested experimentally”, but gives no robustness analysis.
• It is not a sheaf or Kochen–Specker result. X and Z are never measured together. In the record’s own reading (not the paper’s), the cover {X}, {Z} has no overlap, so a global-extension test is vacuous in exactly the scenario where this bound bites.
• It does not say that one binary measurement can witness preparation contextuality. It needs two measurements and four preparations. By the reader’s argument in NOTE-825, one binary measurement never can.
• Satisfying the bound does not establish a noncontextual model. The η-depolarized qubit satisfies it for η ≥ 1 − 1/√2 but is cited as noncontextual only for η ≥ 2/3.
• The three-measurement analogue, |⟨X⟩| + |⟨Y⟩| + |⟨Z⟩| ≤ 1, is verified by computer, not proved in the letter, and is not part of this account.

Connections

• QUESTION-067 (answers, in part). It asks for the equivalences a pragmatic test would assume and the inequality they imply, apart from the sheaf criterion. This is a worked instance: two binary probes, four renderings with equal predictabilities and flipped signs, and one equivalence of equal mixtures. It needs no perfectly distinguishable pairs, which lifts one of the four conditions the question’s note of 2026-10-14 sets. The other three stand: equivalence over every probe (CLAIM-142 and CLAIM-046’s probe-relative relation qualifies only if the probes are tomographically complete for the renderings), the unquotiented description, and exact equivalence.
• THEORY-011. A further qubit-dimension proof of preparation contextuality, without Kochen–Specker structure, which fits its claim that the generalized notion is strictly broader.
• THEORY-015. Compatible. This account concerns preparation noncontextuality alone, and does not use outcome determinism as an assumption.
• THEORY-016. The noncontextual models of the stabilizer and simplicial theories that saturate or escape the bound are instances of its positive representations.
• THEORY-010. Agrees with its point that unpredictability after a change of context has a classical model, and locates what does not: excess joint predictability under the symmetry.
• THEORY-231. That account cites LIT-945 for prepare-measure scenarios admitting noncontextuality no-goes and no Bell-like ones. This is such a no-go, stated as an inequality.
• CLAIM-009, CLAIM-037, CLAIM-044. Their contextuality is the global-extension kind. A violation of this bound would not support them as worded.

Note of 2026-10-19: an independent derivation of the bound

Written for the promotion audit’s batch 019 (ADR-047). It is also in record/promotion-audit/decisions/THEORY-315.yaml.

A preparation-noncontextual model gives preparation s_i a distribution μ_i over ontic states λ. X and Z get response functions ξ_X(λ) and ξ_Z(λ) in [0, 1]. Write a = 2ξ_X − 1 and b = 2ξ_Z − 1, so |a| ≤ 1 and |b| ≤ 1 pointwise, with ⟨X⟩_s = ∫ a dμ_s and ⟨Z⟩_s = ∫ b dμ_s. Treat the μ_i as densities with respect to their sum. By Eq. 10, ⟨X⟩s1 = −⟨X⟩s2 = −⟨X⟩s3 = ⟨X⟩s4 = x and ⟨Z⟩s1 = ⟨Z⟩s2 = −⟨Z⟩s3 = −⟨Z⟩s4 = z.

1. Noncontextuality. Eq. 11 is an operational equivalence, so μ1 + μ3 = μ2 + μ4 =: ν, of total mass 2.
2. Two differences. Put u = μ1 − μ3 and v = μ4 − μ2. The μ_i are non-negative, so |u| ≤ μ1 + μ3 = ν and |v| ≤ μ2 + μ4 = ν pointwise. The second inequality is where step 1 is used.
3. An identity. By Eq. 10, ∫ua = 2x, ∫va = 2x, ∫ub = 2z and ∫vb = −2z. So for any signs σ, τ ∈ {±1}, σx + τz = ¼ ∫ [u(σa + τb) + v(σa − τb)].
4. The bound. Pointwise, |σa + τb| + |σa − τb| = 2 max(|a|, |b|) ≤ 2. So the integrand is at most 2ν, and σx + τz ≤ ¼ · 2 · ∫ν = 1. Choosing σ and τ as the signs of x and z gives |⟨X⟩| + |⟨Z⟩| ≤ 1.

Without step 1, step 2 gives only |v| ≤ μ2 + μ4, and the same steps give 2, the gbit’s value. That shows where noncontextuality does the work. Neither outcome determinism nor any assumption about measurements is used. The paper fixes four outcome-deterministic ontic states. Tightness is the paper’s saturating model (Eqs. A35–A38, checked by hand in NOTE-825).