Source
Budroni, Cabello, Gühne, Kleinmann & Larsson (2022), LIT-263, §IV.B.2, Eqs. 27–28, read in NOTE-844. The construction is credited there to Fritz (2012, Rev. Math. Phys. 24, 1250012), which is not held.
What was actually shown
Treat CHSH as a contextuality scenario whose contexts are the pairs {A_x, B_y}, and call a pair a context when its POVMs are jointly measurable. Joint measurability gives, for each x, y, a POVM G^{xy} with A^x_a = Σ_b G^{xy}{ab} and B^y_b = Σ_a G^{xy}{ab} (Eq. 27). The resulting correlations are automatically nonsignalling. Conversely, for any nonsignalling p(ab|xy), put
G^{xy}{ab} := p(ab|xy)𝟙, A^x_a := Σ_b G^{xy}{ab}, B^y_b := Σ_a G^{xy}_{ab} (Eq. 28),
on a one-dimensional Hilbert space. Nonsignalling is exactly what makes A^x_a independent of y and B^y_b independent of x. So every pair is jointly measurable, and every nonsignalling behaviour, the PR box included, is reproduced. The review states that the argument “extends straightforwardly to any arbitrary contextuality scenario”. The nondisturbing correlations replace the nonsignalling ones, and with them “all maximally contextual correlations”.
The general scenario (checked in NOTE-844 and re-derived for the promotion audit, ADR-047 v2). A marginal scenario has observables X, each with a finite outcome set O_X, and a cover of contexts C. A behaviour gives a distribution p_C on O_C = Π_{X∈C} O_X for each context, and it is nondisturbing when p_C and p_C′ have the same marginal on C ∩ C′.
- Take the one-dimensional Hilbert space ℂ and its only state, ρ = 1.
- For each context C and each a_C ∈ O_C, put G^C_{a_C} := p_C(a_C)𝟙. Each effect is a non-negative multiple of 𝟙, and they sum to 𝟙, so G^C is a POVM on O_C.
- For X ∈ C, the induced POVM is E^{X,C}a := Σ{a_C : a_C(X) = a} G^C_{a_C} = p_C|_X(a)𝟙.
- Nondisturbance on {X} ⊆ C ∩ C′ gives p_C|_X = p_C′|_X. So E^{X,C} does not depend on C: each observable is a single POVM E^X, and each context’s observables are jointly measurable, with G^C as the joint POVM.
- Born’s rule gives tr(ρ G^C_{a_C}) = p_C(a_C), so every context’s statistics are reproduced. That includes behaviours with no global section, such as the PR box and the maximally contextual behaviours.
So if a context is any jointly measurable set, every nondisturbing behaviour has a quantum realization by effects proportional to the identity, and contextuality relative to that notion of context marks nothing.
The review concludes that “when one uses too broad of a notion of contexts, i.e., joint-measurability alone, contextuality becomes a trivial property”. It adopts contexts made of sequences of outcome-repeatable, statistically nondisturbing measurements (§IV.B.2, S.1–S.5).
What could have failed: some nondisturbing behaviour might have needed genuinely incompatible effects to be realized as jointly measurable families. None does.
What this does not say
- Not that joint measurability is irrelevant. It is “the minimal requirement to define some notion of a context” (§II.B). The claim is that it is not sufficient.
- Not that contextuality is trivial. For sharp measurements, commutativity, joint measurability, nondisturbance and repeatability coincide (§II.B, §IV.B.2), and Kochen–Specker contextuality is not trivial. The triviality comes from admitting unsharp, here maximally unsharp, effects without further constraint.
- Not the only escape. Liang, Spekkens & Wiseman (2011) keep joint measurability and add a sharpness parameter that raises the noncontextual bound, to the algebraic maximum for effects proportional to the identity (§IV.B.2). Henson & Sainz (2015) and Kunjwal (2019) are cited for other views. Spekkens’s generalized contextuality sidesteps the issue by identifying effects by their statistics.
- Nothing about which operational criterion is right outside physics. The review’s criterion, sequential repeatability and nondisturbance, is built for laboratory measurements.
Connections
- THEORY-012 defines noncontextuality as gluing context distributions to a global one, with the contexts given. This account says the definition has empirical content only once the contexts are fixed by something stronger than joint measurability.
- QUESTION-067 asks which noncontextuality assumption a test of pragmatic judgements would refute. This account adds a precondition: whichever is chosen, what makes two renderings “the same measurement in two contexts” cannot rest on joint performability alone.
- THEORY-303, from the same reading, gives the structural side. Even with contexts fixed, an acyclic compatibility structure cannot be contextual.