NOTE-825: Catani et al. — What is nonclassical about uncertainty relations?

Contribution

Before this letter it was known that a nontrivial trade-off between the predictabilities of two measurements occurs in theories that admit a generalized-noncontextual model (Gaussian quantum mechanics, odd-prime stabilizer theories, the qubit stabilizer theory), so the mere existence of an uncertainty relation is not evidence of nonclassicality. The letter derives a noncontextuality inequality, |⟨X⟩| + |⟨Z⟩| ≤ 1, that bounds the predictabilities of two binary measurements for any state with a certain symmetry orbit. In theories where every state has that orbit it is a bound on the form of the uncertainty relation, and the qubit’s circular trade-off violates it.

Key insight

Noncontextuality cannot constrain a single state. It constrains states through the operational equivalences among them. The trick is that a state’s predictabilities are shared by its reflections: the four states with the same |⟨X⟩| and |⟨Z⟩| and all four sign patterns. When those four also satisfy one equivalence of mixtures, the predictabilities of any one of them are bounded by a linear diamond. So what is nonclassical about the qubit is not that X and Z cannot both be predictable, but that they can be more jointly predictable than any noncontextual theory with the same symmetry allows.

Assumptions

• Prepare-measure scenarios only, with a finite ontic state space (“it suffices to consider Λ finite”).
• Generalized noncontextuality for preparations only (footnote 3): P ≃ P′ ⇒ μ(λ|P) = μ(λ|P′), where P ≃ P′ means “leading to the same statistics for all possible measurements” (Eq. 8). The term is used throughout in Spekkens’s sense, “distinct from the Kochen-Specker notion” (footnote 1).
• Binary measurements X and Z, with predictability |⟨X⟩|, where ⟨X⟩ = (e₊₁|X − e₋₁|X)·s (Eq. A1).
• The A²₁-orbit-realizability condition on a state s1, which is two conditions:
  • equal-predictability counterparts (Eq. 10): states s2, s3, s4 with ⟨X⟩s1 = −⟨X⟩s2 = −⟨X⟩s3 = ⟨X⟩s4 and ⟨Z⟩s1 = ⟨Z⟩s2 = −⟨Z⟩s3 = −⟨Z⟩s4;
  • an operational equivalence (Eq. 11): ½s1 + ½s3 = ½s2 + ½s4. For two measurements the second does not follow from the first and must be imposed (Appendix C).
• Outcome-deterministic response functions, justified, not assumed: following Schmid, Spekkens and Wolfe, determinism is without loss of generality when there are no nontrivial operational equivalences among the measurement effects, “as is the case here” (Appendix A).
• Exact operational equivalence. The proof uses Eq. 11 as an equality. The text says the inequality “is noise-robust, and can therefore be tested experimentally” with the techniques of Mazurek et al., but it does not carry out that analysis.
• A 4-dimensional real vector representation for all the foil theories compared. The general result does not need it.

Key results

• Main result (Eq. 12; Appendix A). In any operational theory, if a state s1 is A²₁-orbit-realizable relative to X and Z, then any preparation-noncontextual model of these states and measurements satisfies |⟨X⟩| + |⟨Z⟩| ≤ 1.
  • The proof reduces to four ontic states (one per pair of X and Z outcomes), parameterizes μ2, μ3, μ4 from μ1 = (a, b, c, d) by single parameters ε, γ, δ (Eqs. A13–A27), imposes Eq. A6 (½μ1 + ½μ3 = ½μ2 + ½μ4, from Eq. 11 and noncontextuality) to get a + d = b + c + (ε + γ + δ), and bounds ⟨X⟩ + ⟨Z⟩ = 2(b − c) = 1 − (ε + γ + δ) − 4c ≤ 1 − c ≤ 1 (Eq. A33).
  • Equivalently four linear inequalities, ±⟨X⟩ ± ⟨Z⟩ ≤ 1 (Eq. A34), a diamond (Fig. 4).
  • Tight: the explicit model μ1 = (1/4 + u, 1/2, 0, 1/4 − u) and its reflections (Eqs. A35–A38), u ∈ [−1/4, 1/4], reaches every point with ⟨X⟩ + ⟨Z⟩ = 1.
  • It also follows from the complete set of eight inequalities that Schmid, Spekkens and Wolfe derived by linear program for this equivalence (Appendix B, Eq. B1), which reduce to Eq. A34 under the A²₁ condition. The letter corrects a typo in that earlier paper (P12, not P21, in the sixth inequality).
  • Holds when: the orbit condition holds for the state. It needs no perfectly distinguishable pairs.
• Constraint on the form of the uncertainty relation. A theory in which every state is A²₁-orbit-realizable is said to have A²₁ symmetry. There Eq. 12 is “a universal constraint on the predictability tradeoff”, that is, on the ZX-uncertainty relation itself.
• The qubit violates it. The qubit’s ZX-uncertainty relation is ⟨X⟩² + ⟨Z⟩² ≤ 1 (Eq. 1), the strongest one for X and Z (Appendix D, where it is also written ΔX² + ΔZ² ≥ 1 and Cx² + Cz² ≤ 3/2). It meets the bound only where |⟨X⟩| = 1 or |⟨Z⟩| = 1, and the largest violation is |⟨X⟩| + |⟨Z⟩| = √2 at |⟨X⟩| = |⟨Z⟩| = 1/√2.
• The foils (Eqs. 3–6; Fig. 2a).
  • Qubit stabilizer theory (octahedron): |⟨X⟩| + |⟨Z⟩| ≤ 1, which saturates the bound; it admits a noncontextual model.
  • η-depolarized qubit (ball of radius 1 − η): ⟨X⟩² + ⟨Z⟩² ≤ (1 − η)². It satisfies the bound if η ≥ 1 − 1/√2 ≈ 0.293. It admits a noncontextual model for η ≥ 2/3 (Marvian, cited).
  • Gbit (cube): |⟨X⟩|, |⟨Z⟩| ≤ 1, with A²₁ symmetry, so it reaches the algebraic maximum 2 and is “even further than the qubit theory from being classically explainable”.
  • Simplicial theory (tetrahedron, two classical bits): the same relation as the gbit, but it does not have A²₁ symmetry. Its orbit- realizable states are exactly the embedded octahedron, which satisfies the bound. The vertices that reach |⟨X⟩| = |⟨Z⟩| = 1 have equal-predictability counterparts but fail Eq. 11. So no contextuality is witnessed, as it must not be, since this theory has a noncontextual model (the vertices as ontic states).
• Three measurements (Eq. 13; Appendix C). For an A³₁-orbit- realizable state, |⟨X⟩| + |⟨Y⟩| + |⟨Z⟩| ≤ 1 (an octahedron, Fig. 7). Here the equivalences (Eq. C6) follow from the equal-predictability condition alone, so the orbit condition is just Eq. C5. The qubit reaches √3 at |⟨X⟩| = |⟨Y⟩| = |⟨Z⟩| = 1/√3. Not proved analytically: “we do not provide an analytic proof here”; it was checked by computational algebra (acknowledgement to Elie Wolfe).
• A remark on axioms (Appendix C). Taking A³₁ symmetry plus “one can realize all states consistent with noncontextuality” as axioms for a 4-dimensional real representation yields the octahedral state space, the state space of Spekkens’s toy theory. Stated as a suggestion, not developed.

Claims

id	claim	strength	support
C1	Existence of an uncertainty relation is not evidence of nonclassicality	strong	prior results cited (Gaussian QM, odd-d stabilizer), and the qubit stabilizer theory exhibited here
C2	For an A²₁-orbit-realizable state, preparation noncontextuality implies |⟨X⟩| + |⟨Z⟩| ≤ 1, and the bound is tight	strong	analytic proof and explicit saturating model, Appendix A; LP cross-check, Appendix B
C3	In A²₁-symmetric theories the form of the ZX-uncertainty relation witnesses contextuality; the qubit’s does	strong	C2 plus the qubit’s A²₁ symmetry, shown geometrically (Fig. 3)
C4	Nonclassicality grows with joint predictability, not with unpredictability	moderate	an interpretation of C2–C3 within A²₁-symmetric theories; the simplicial theory shows it fails without the symmetry
C5	The three-measurement bound |⟨X⟩| + |⟨Y⟩| + |⟨Z⟩| ≤ 1 holds	moderate	computational verification only, Appendix C
C6	Eq. 12 is noise-robust and experimentally testable	weak	asserted with references; no robustness analysis in the letter

Method

A noncontextuality inequality for a fixed prepare-measure scenario (four preparations, two binary measurements, one equivalence among mixtures), derived by hand: reduce the ontic space to the four outcome-deterministic assignments, write each μi from μ1 and one free parameter using the equal-predictability constraints, impose the equivalence on the μi, and eliminate the parameters. Then reinterpret the inequality, under a symmetry assumption, as a bound on the uncertainty relation of a whole theory.

Concepts

• Predictability. |⟨X⟩|, the absolute expectation of a ±1-valued binary measurement. Variance is 1 − ⟨X⟩² (Appendix D).
• Equal-predictability counterparts. Four states with the same |⟨X⟩|, |⟨Z⟩| and all four sign patterns (Eq. 10).
• A²₁-orbit-realizability. A state has equal-predictability counterparts that also satisfy ½s1 + ½s3 = ½s2 + ½s4 (Eqs. 10–11). The four form a rectangle parallel to the x̂ẑ-plane, the orbit of s1 under the reflection group of a rectangle.
• A²₁ symmetry (of a theory). Every state is A²₁-orbit-realizable relative to the chosen X and Z.
• Foil theory. An operational theory studied for its contrast with quantum theory.
• Noncontextual (here). Generalized noncontextuality for preparations, not Kochen–Specker noncontextuality.

Connections

The definition is Spekkens’s (2005, LIT-266, read in NOTE-748). The eight-inequality polytope it specializes is Schmid, Spekkens and Wolfe’s (PRA 97, 062103, 2018), and the simplest scenario with a nontrivial equivalence, four states, is Pusey’s (PRA 98, 022112, 2018). Neither is held here. The noncontextual model of the simplicial theory is cited from Schmid et al. 2021 (LIT-003). The program it belongs to, asking which aspects of a phenomenon resist a noncontextual model, is the one stated in Catani, Leifer, Schmid and Spekkens (LIT-019), which the letter cites as [6]. The octahedral state space is that of Spekkens’s toy theory and its epistemically restricted generalizations (LIT-007 is one).

Bearing on the record

• QUESTION-067: the filing claim is confirmed, with one precision. The question’s status note says the paper shows uncertainty-type trade-offs compatible with generalized noncontextuality. The letter cites that result (Gaussian QM, odd-prime stabilizer theories) and exhibits one more case (the qubit stabilizer theory). It proves the other half: the form of a trade-off can violate noncontextuality.
• For QUESTION-067’s “What would answer it”. The question asks for the operational equivalences a pragmatic experiment would assume and the inequality they imply, separate from the sheaf criterion. This letter is a worked template. Four renderings with equal predictabilities on two binary probes and flipped signs, plus one equivalence of equal mixtures, give |⟨X⟩| + |⟨Z⟩| ≤ 1. The two probes are never made together, so in the record’s own join (not the paper’s), the measurement cover is {X}, {Z} with no overlap, and a sheaf or Kochen–Specker test is vacuous there. Filed as THEORY-315.
• On QUESTION-067’s four conditions (note of 2026-10-14).
  • Exact equivalence: still needed. Eq. 11 is used as an equality, and the claimed noise robustness is asserted, not shown.
  • Perfectly distinguishable decompositions: not needed here. The note says a pragmatic version “needs such pairs or a noise-robust inequality, which neither reading supplies”. This letter supplies an inequality with no distinguishability condition; s1 and s3 can be any states with the orbit. The note should name it.
  • Equivalence over every probe: still needed. Eq. 8 quantifies over “all possible measurements”.
• QUESTION-067’s “single binary judgement”. The question’s “Why the record needs it” says preparation contextuality “can show even with a single binary judgement”. This letter does not say that, and its witness needs two binary measurements and four preparations. The letter’s own scenario is the simplest it knows, citing Pusey. On the reader’s argument (not the paper’s), data from one binary measurement always admit a preparation-noncontextual model: take the two outcomes as ontic states and μ(+|P) = P(+|M, P), which depends on P only through its statistics and respects mixtures. THEORY-231 and NOTE-737 repeat the phrase as supported by LIT-945, which shows only that a prepare-measure scenario has no Bell-like test. The phrase should read “two binary judgements never made together”.
• THEORY-011 and THEORY-015. Consistent. The letter is another qubit-dimension preparation-contextuality proof, one that needs no perfectly distinguishable pairs and no Kochen–Specker structure.
• THEORY-016. Its noncontextual models of the stabilizer and simplicial theories are the ones that saturate or escape the bound. Nothing here extends it.
• THEORY-010. It says unpredictability after a change of context is reproduced by an epistemically restricted model. This letter agrees and sharpens it: the qubit stabilizer (octahedral) theory has a nontrivial trade-off and a noncontextual model. What resists such a model is too much joint predictability, given A²₁ symmetry.
• CLAIM-009 and CLAIM-037. Both are stated in the global-extension sense. A violation of Eq. 12 is not a failure of global extension, so it would not support either as worded. It would support a preparation version that neither states.
• No instruction for machine-learning practice; nothing for the anthology.

Limitations

• The symmetry does the work. Outside A²₁-symmetric theories the bound covers only the orbit-realizable states, and a theory can have any trade-off at all, as the simplicial theory shows.
• Exact equivalence. The proof assumes Eq. 11 exactly. No robustness analysis is given.
• The three-measurement bound is verified by computer, not proved.
• A gap left open. The η-depolarized qubit satisfies the bound for η ≥ 0.293 but is cited as having a noncontextual model only for η ≥ 2/3. Satisfying this one inequality is necessary, not sufficient, and the letter does not discuss 0.293 ≤ η < 2/3.
• Typos. Appendix C says the depolarized qubit satisfies the three-measurement bound “if η ≤ 1 − 1/√3”; the geometry (radius 1 − η, maximum √3(1 − η)) gives η ≥. Appendix A refers to “Fig. 7” for the saturating distributions, which are in Fig. 5.

Open questions

• Does the bound survive inexact equivalences, and with what correction term? A robustness analysis in the style of Mazurek et al. would answer it.
• For the depolarized qubit with 0.293 ≤ η < 2/3, is there a noncontextual model, or another inequality that the bound misses?
• Does the axiomatic remark (A³₁ symmetry plus maximal noncontextual state space gives the octahedron) generalize past dimension 4?