Source
Spekkens (2005), LIT-266, §III, §VIII.A–B, §IX (NOTE-748). Supporting, thinly: Budroni et al. (LIT-263, §IV.B.3 on unsharp response functions under measurement noncontextuality, §IV.E.1 for the definitions, and §IV.E.4 on the preparation contextuality of Bell’s and Kochen & Specker’s qubit models; the review does not state the decomposition or the implication, NOTE-844); Abramsky & Brandenburger, LIT-016, Thm 8.1 and Prop 3.1; Spekkens (2014), LIT-007, §IV.B.
What was actually shown
Spekkens shows that the traditional (Kochen–Specker) notion is measurement noncontextuality — independence of a sharp measurement’s representation from how its projectors are grouped — together with outcome determinism, sharp measurements being represented by indicator functions (§III). He then proves that preparation noncontextuality forces outcome determinism for every projective measurement: the ontic supports of an orthonormal basis’s preparations must cover the ontic space, because they all decompose the maximally mixed state, whose representation is unique (§VIII.A, Eqs. 77–88; the proof is given for rank-1 projective measurements and extended to the rest by coarse-graining). So universal noncontextuality implies Kochen–Specker noncontextuality, and every Kochen–Specker proof is a proof against the universal notion. Measurement noncontextuality on its own is consistent with quantum theory: the Beltrametti–Bugajski model is measurement-noncontextual for all POVMs, and pays with outcome indeterminism and preparation contextuality (§VIII.B). “One cannot confine all the contextuality into the measurements” (§IX).
The held record already uses the link in one direction: Spekkens 2014 rules out a nonnegative quasiprobability representation of the qubit stabilizer theory because that theory contains the Mermin-square and GHZ Kochen–Specker proofs (LIT-007 §IV.B, by citation).
The record’s own join: by LIT-016 Thm 8.1 and Prop 3.1, a sheaf-contextual model built from projective measurements has no outcome-deterministic, measurement-noncontextual model; by the above, the operational theory it comes from then has no universally noncontextual model — provided it contains the preparations §VIII.A uses.
What this does not say
- That the two notions coincide (they do not: see the qubit claim in this group).
- That Kochen–Specker proofs refute preparation noncontextuality alone. They refute preparation plus measurement noncontextuality, which universal noncontextuality includes; refuting preparation noncontextuality alone takes a proof such as the one in §IV (THEORY-011).
- That the implication holds in restricted fragments. §VIII.A uses quantum theory’s full preparation set (Eqs. 85–86); inside a stabilizer or interferometric fragment (LIT-019) it must be re-checked.
- Anything about noisy or unsharp Kochen–Specker tests, which is Kunjwal & Spekkens’s programme (not held, not read).
- That the sheaf-form statement appears in any source; it is the record’s join.