Samson Abramsky, Rui Soares Barbosa, Shane Mansfield (2017), Physical Review Letters 119, 050504 (2017) — ARXIV-1705.07918
Key takeaways
- Definition. For an empirical model e on ⟨X, M, O⟩, with marginals agreeing on overlaps, NCF(e) is the largest λ in e = λe^NC + (1 − λ)e′ with e^NC noncontextual and e′ an empirical model. Equivalently, it is the largest weight of a subprobability b on global assignments with b|_C ≤ e_C. CF = 1 − NCF. Every model splits as NCF·e^NC + CF·e^SC with e^SC strongly contextual, not uniquely: the supplement’s Table II gives two such splits of one (3,2,2) model. CF = 1 is strong contextuality (LIT-016).
- Linear programming and Bell inequalities (Eqs. 3–4, Theorem 1). NCF = max 1·b subject to Mb ≤ v_e, b ≥ 0. With y* solving the dual (min y·v_e, Mᵀy ≥ 1, y ≥ 0), a* = |M|⁻¹1 − y* defines a Bell inequality with bound 0 and algebraic bound 1. Its violation by e is exactly CF(e), and no Bell inequality’s normalised violation exceeds CF(e). It is tight at e^NC and maximally violated by e^SC in every optimal decomposition.
- Monotonicity (Theorem 2). CF is invariant under relabelling and non-increasing under translation of measurements (including restriction to fewer measurements) and coarse-graining of outcomes. It is convex under mixing. CF(e & e′) = max{CF(e), CF(e′)}, and NCF(e ⊗ e′) = NCF(e)·NCF(e′). The mixing bullet of the printed theorem has a typo (CF(e) for CF(e′)); the proof is for the correct statement.
- Computational advantage (Theorems 3–4). An l2-MBQC computing f with average failure p̄_F has p̄_F ≥ NCF(e)·ν̃(f), where ν̃(f) is f’s average distance to the nearest Z2-linear function. A valid strategy for a k-consistent system of n formulae has p_F ≥ NCF·(n − k)/n.
- Examples (supplement B). On |Φ⁺⟩ with equatorial measurements, CF peaks at the Tsirelson models, not at the textbook CHSH angles, and is never 1. On GHZ(n), CF = 1 at a family of angle pairs that are relabellings of Mermin’s X/Y model (Proposition 4).
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
Deferred then because only the Letter had been skimmed. Re-filed on
2026-10-09 from the manuscript bibliography of 2026-10-09 (work
what-survives-translation) at the owner’s request; see the curation entry
of that day.
Read on 2026-10-09 (NOTE-236) and placed: Active. The supplemental proofs that the 2026-09-27 seed left unchecked (Theorem 1(ii)–(iii), the Theorem 3 conditions) are complete and correct, apart from the typo above. The reading is the source of THEORY-165: within a scenario, CF is a convex, piecewise-linear, Lipschitz function of the probability table. That derivation is the reader’s, from the paper’s linear programme. The contrast between CF’s monotonicity under coarse-graining and the behaviour of Contextuality-by-Default under coarse-graining is drawn in THEORY-168. CF is not defined for data with context-dependent marginals, which is where Contextuality-by-Default (LIT-777) applies. The paper leaves the comparison of the two measures to future work (its note 33).