Contribution
LIT-016 had the non-contextual fraction only at its extreme value: zero non-contextual fraction is strong contextuality. This paper makes the fraction a graded measure for any scenario. It shows that the measure is a linear programme, that it equals the best normalised Bell-inequality violation (with the inequality computed by the dual), that it behaves as a monotone under a set of operations that cannot create contextuality, and that it bounds quantum computational advantage in two settings.
Key insight
Ask what fraction of the data a noncontextual model can explain from below: the heaviest subprobability on global assignments that fits under every context’s distribution. That is a linear programme with the data on the right-hand side, and LP duality turns it into the statement that the unexplained remainder is exactly how far the data violate the best Bell inequality. One number therefore does three jobs: a decomposition weight, a violation, and a resource.
Assumptions
- A finite measurement scenario ⟨X, M, O⟩: finite measurements, finite outcomes, maximal contexts M covering X.
- Empirical models with compatible marginals: e_C|{C∩C′} = e{C′}|_{C∩C′} (“a generalisation of the usual no-signalling condition”). Data with context-dependent marginals are not empirical models, and CF is undefined for them.
- Bell inequalities ⟨a, R⟩ with R < ‖a‖ (the algebraic bound), so the normalisation is defined.
- For Theorem 3: (n, 2, 2) resources, Z2-linear pre- and post-processing and a strictly lower-triangular flow matrix. For Theorem 4: uniformly random formulae.
Key results
- Eq. (1)–(2). NCF(e) = max λ over e = λe^NC + (1 − λ)e′. Every model decomposes as NCF·e^NC + CF·e^SC with e^SC strongly contextual (credited to LIT-016), not uniquely. The supplement’s Table II gives two decompositions of one model with CF = ½ in the (3,2,2) scenario. Uniqueness holds when the no-signalling polytope has no face made only of strongly contextual vertices, as in (2,2,2).
- LP (3) and dual (4). NCF = max 1·b subject to Mb ≤ v_e, b ≥ 0. Dual: min y·v_e subject to Mᵀy ≥ 1, y ≥ 0. Strong duality applies, because the primal is feasible (b = 0) and bounded.
- Theorem 1. (i) Every Bell inequality’s normalised violation max{0, a·v_e − R}/(‖a‖ − R) is at most CF(e). The proof splits e into NCF·e^NC + CF·e^SC and bounds each part. (ii) If CF(e) > 0, then a* = |M|⁻¹1 − y* with bound 0 is a Bell inequality with algebraic bound 1, violated by exactly CF(e). The change of variables makes the dual equivalent to max a·v_e subject to Mᵀa ≤ 0, a ≤ |M|⁻¹1. (iii) In any optimal decomposition, that inequality is tight at e^NC and maximally violated at e
- Theorem 2. CF is invariant under relabelling. It is non-increasing under translation of measurements (pulling e back along a context-preserving f: X → X′, which includes restriction) and under coarse-graining of outcomes. NCF(λe₁ + (1 − λ)e₂) ≥ λNCF(e₁) + (1 − λ)NCF(e₂); the printed CF form has a typo. CF(e & e′) = max{CF(e), CF(e′)}, using Lemmas 5–6, which glue two subprobabilities of equal weight into a joint one. NCF(e ⊗ e′) = NCF(e)·NCF(e′), by product primal and product dual solutions.
- Theorem 3. For an l2-MBQC ⟨K, e⟩ computing f: 2^m → 2^l, p̄_F ≥ NCF(e)·ν̃(f). Proof: split e. Each deterministic noncontextual part computes a Z2-linear function (Raussendorf), so it fails on at least a ν̃(f) fraction of inputs.
- Theorem 4. For a k-consistent set of n formulae and a valid strategy p, p_F ≥ NCF(p)·(n − k)/n, from Theorem 1 and Abramsky–Hardy’s logical Bell inequalities.
- Proposition 4. Equatorial measurements at (φ₁, φ₂) = ((n + k)π/2n, kπ/2n), 0 ≤ k < n, on GHZ(n) give Mermin’s strongly contextual model up to relabelling.
- Computational findings (supplement B). On |Φ⁺⟩ with two equatorial settings per qubit, CF depends on the absolute angles, not only on their difference. Its maxima are the Tsirelson-violating models, none strongly contextual. On GHZ(3) and GHZ(4), CF = 1 at the angles of Proposition 4.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | CF(e) equals the largest normalised violation of any Bell inequality by e, attained by an inequality computed from the dual LP | strong (proof) | Theorem 1, supplement A |
| C2 | CF is invariant under relabelling and non-increasing under translation of measurements and coarse-graining of outcomes | strong (proof) | Theorem 2, supplement C |
| C3 | CF is convex under mixing, max under choice, and NCF is multiplicative under product | strong (proof) | Theorem 2, supplement C, Lemmas 5–6 |
| C4 | The noncontextual / strongly contextual decomposition is not unique in general | strong (example) | Table II |
| C5 | Z2-linear MBQC failure probability is at least NCF·ν̃(f) | strong (proof, relying on Raussendorf’s linearity lemma) | Theorem 3, supplement D |
| C6 | k-consistent games: failure at least NCF·(n − k)/n | strong (proof via Abramsky–Hardy) | Theorem 4 |
| C7 | CF allows meaningful comparison across scenarios | moderate (normalisation to [0, 1] and C1, not a theorem about maps between scenarios) | Introduction |
Method
Everything is linear programming over the incidence matrix M, whose columns are the deterministic noncontextual models. The primal finds the heaviest noncontextual subprobability under the data. The dual, after a change of variables, finds the most-violated normalised Bell inequality. The monotonicity proofs transport an optimal primal solution, or for products also an optimal dual one, along each operation.
Concepts
- empirical model. A family of distributions e_C on O^C, one per maximal context, with compatible marginals.
- non-contextual / contextual fraction. NCF(e) is the largest weight of a subprobability b on O^X with b|_C ≤ e_C. CF = 1 − NCF.
- normalised violation. max{0, a·v_e − R}/(‖a‖ − R), where ‖a‖ is the sum over contexts of the largest coefficient.
- free operations. Relabelling, translation of measurements, coarse-graining, mixing, controlled choice (&) and product (⊗), named as the start of a resource theory to be given elsewhere.
- ν̃(f). The average Hamming-style distance from f to the nearest Z2-linear function.
Connections
It is built on LIT-016’s framework, and the decomposition into a strongly contextual part is from there. It generalises the Elitzur–Popescu–Rohrlich local fraction from Bell scenarios to all scenarios. Theorem 3 sharpens Raussendorf (2013); Theorem 4 uses Abramsky and Hardy’s logical Bell inequalities. Its note 33 lists negative-probability measures (LIT-016), Contextuality-by-Default measures and noise and inefficiency measures as alternatives whose relation to CF is left for later.
Bearing on the record
- THEORY-012. The theory says the graded version of its criterion is this paper’s. The reading confirms it: CF = 0 exactly when the global-section criterion holds, and the LP is the relaxation of LIT-016’s linear system. The inactive-ok directives citing LIT-265 as Deferred, in THEORY-012, LIT-777, NOTE-600 and CLAIM-105, are now stale.
- THEORY-014. The negative-probability measure appears only in note 33. The paper’s measure is the nonnegative side of that theory’s pattern, the largest nonnegative part, rather than the least negative completion. It adds no instance and does not contradict it.
- THEORY-165 is filed from this reading. Its convexity is Theorem 2. Its piecewise linearity and Lipschitz continuity in v_e are the reader’s derivation from LP (3)–(4): NCF(e) = min over the finitely many vertices y_k of the dual polyhedron of y_k·v_e.
- THEORY-168. CF’s non-increase under coarse-graining (C2) contrasts with Contextuality-by-Default, where coarse-graining a non-binary system can create contextuality (LIT-831, Example 3). That theory states the contrast.
- The manuscript (CLAIM-105, Δ_CF). Three bearings, for the coordinator: (1) Δ_CF is defined only when both models in a step have compatible marginals. A reconstruction that introduces context-dependent marginals leaves CF undefined. (2) By Theorem 2, a step that is a free operation (relabelling, restriction or translation of measurements, coarse-graining of outcomes, mixing with a noncontextual model) cannot increase CF. Along such steps, Δ_CF ≤ 0, and an increase in contextuality needs a step outside the free operations. This supports “local meanings preserved, global structure changed” in the decreasing direction and constrains the increasing one. (3) The claim’s quoted premise that CF is “a discontinuous contextuality measure” is not borne out within a scenario. CF is continuous, indeed Lipschitz, in the probability table (THEORY-165, the reader’s derivation). A drift bound on every context’s distribution therefore bounds |Δ_CF| within a fixed scenario. What may jump is a change of scenario (cover), which is not a perturbation of the table.
- No instruction for machine-learning practice. Theorem 3 concerns quantum computation, not ML.
Limitations
- Defined only for no-signalling (compatible-marginal) models.
- The resource theory is announced, not given. “Free operations” are a list, not a characterization.
- C7, comparison across scenarios, is motivated by normalisation, not proved for any map between scenarios beyond translations.
- Theorem 3 holds only for Z2-linear classical control and (n, 2, 2) resources.
- The decomposition is not unique, so e^NC and e^SC are not invariants of the model.
Open questions
- The relation between CF and Contextuality-by-Default’s total-variation measure (LIT-777) on consistently connected systems, which the paper defers. On those systems both are LPs over closely related polytopes, and whether they order models the same way is not known to the record.
- An explicit Lipschitz constant for CF in a given scenario, the largest ‖y‖₁ over dual vertices. A bound would turn drift bounds on context distributions into bounds on Δ_CF.