NOTE-748: Spekkens — Contextuality for preparations, transformations, and unsharp measurements

Contribution

The paper gives a definition of noncontextuality that applies to any operational theory, to every kind of procedure, and to ontological models whose outcomes are only probabilistically fixed by the ontic state. It then proves three no-go theorems in a two-dimensional Hilbert space: one each for preparations, unsharp measurements and transformations. It shows how the traditional Kochen–Specker notion sits inside the new one. Before it, contextuality was a property of sharp measurements in dimension three or more. After it, it is a property of the representation of any procedure, and a qubit already has it.

Key insight

A context is whatever about a procedure is not fixed by its operational equivalence class: “A noncontextual ontological model of an operational theory is one wherein if two experimental procedures are operationally equivalent, then they have equivalent representations in the ontological model.” Wherever the operational formalism has a freedom of decomposition, there is a context, and “one can develop a proof of contextuality that is based on this freedom”. Mixed states have many convex decompositions even for a qubit, so preparation contextuality needs no third dimension and no assumption about measurements.

Assumptions

• Operational equivalence quantifies over every procedure the theory allows. Preparations P and P′ are equivalent if p(k|P, M) = p(k|P′, M) for all M (Eq. 1); measurements likewise over all P (Eq. 2); transformations over all P and M (Eq. 3).
• The ontological model framework (Eq. 4). Each preparation is a normalised density µ_P(λ); each measurement a set of indicator functions ξ_{M,k}(λ) ∈ [0,1] with Σ_k ξ_{M,k}(λ) = 1; each transformation a stochastic transition matrix Γ_T(λ′, λ); and p(k|P,T,M) = ∫dλ′dλ ξ_{M,k}(λ′) Γ_T(λ′,λ) µ_P(λ) reproduces the theory exactly.
• Feature 1. Preparations distinguishable with certainty in one shot have nonoverlapping distributions: µ(λ)µ′(λ) = 0 (Eq. 11).
• Feature 2. A convex mixture of procedures is represented by the same convex mixture of representations (Eq. 12, and its analogues for measurements and transformations).
• Noncontextuality, per kind. µ_P = µ_{e(P)} (Eq. 5), ξ_{M,k} = ξ_{e(M),k} (Eq. 6), Γ_T = Γ_{e(T)} (Eq. 7), where e(·) is the equivalence class. “Universally noncontextual” means all three.
• Outcome determinism for a measurement means idempotent indicator functions, χ(λ)² = χ(λ). The §V proofs assume it for sharp measurements only; the paper says so: those “proofs of contextuality” “are contingent on the assumption of outcome determinism for sharp measurements”.
• §VIII.A uses the full set of quantum preparations. The ontic space is defined as Ω = {λ | µ_ρ(λ) > 0 for some ρ} (Eq. 85), and the proof needs that “every density operator ρ appears in some convex decomposition of the completely mixed state I/d”.
• Exact equivalence. The definitions apply only when two procedures are exactly operationally equivalent; §IX concedes that finite precision may mean this never happens in practice.

Key results

• Traditional noncontextuality as a special case (§III). For sharp measurements with outcome determinism, independence of the representation from the fine-graining of the PVM “is just the traditional notion of noncontextuality”. Measuring A with B or with C is a choice of fine-graining of A’s PVM. Traditional proofs need dimension ≥ 3 “because there are no fine-grainings of non-trivial PVM measurements in a 2d Hilbert space”.
• Preparation contextuality for a qubit (§IV, Eqs. 13–36). Six pure states a, A, b, B, c, C at 60° intervals on a great circle of the Bloch sphere give five decompositions of I/2: ½(a + A), ½(b + B), ½(c + C), ⅓(a + b + c), ⅓(A + B + C) (Eqs. 19–23). Orthogonal pairs give µ_aµ_A = µ_bµ_B = µ_cµ_C = 0 (Eqs. 24–26). Preparation noncontextuality sets all five mixtures equal to one ν(λ) (Eqs. 32–36). At each λ, the eight ways of zeroing one of each pair all force the all-zero solution. Two cases are worked; the other six follow by 60° and 120° rotations, which I checked (the three mixed cases of each kind are rotations of one another, and a 60° rotation swaps the triangles). No normalised solution exists. The proof uses no measurement assumption beyond Feature 1. (The text defines P_abc as a mixture of P_a, P_b and P_C; Eq. 22 shows that P_c is meant.)
• Unsharp-measurement contextuality (§V.A, Eqs. 37–52). Mixing the three sharp measurements {P_a, P_A}, {P_b, P_B}, {P_c, P_C} with weight ⅓ gives the POVM {½I, ½I}, operationally a fair coin, so measurement noncontextuality demands indicator functions {½, ½}. Outcome determinism for the sharp ones allows only {0,1}, {1,0}, {⅔,⅓} or {⅓,⅔}. Contradiction.
• Gleason-type version (§V.B, Eq. 53). Measurement noncontextuality for all POVMs plus Busch’s and Caves et al.’s 2d Gleason theorem give ξ_E(λ) = Tr(ρ_λ E). Then χ_P(λ) = Tr(ρ_λ P) cannot be idempotent for every projector.
• Transformation contextuality for a qubit (§VI, Eqs. 54–66; App. A). Six rotations about the y axis by multiples of π/3 give five convex decompositions of the map that projects the Bloch ball onto the y axis. Transformation noncontextuality then reproduces the §IV constraints on the images of µ_a. Appendix A proves the decompositions by unitary remixing of Kraus operators (Eqs. A4, A5). Footnote 11: by the Jamiołkowski isomorphism this is a preparation-contextuality proof in four dimensions. The proof is due to Rudolph.
• Outcome determinism for unsharp measurements is excluded outright (§VIII). The POVM {I/2, I/2} must be represented by {1/2, 1/2}, which is not idempotent.
• Preparation noncontextuality implies outcome determinism for sharp measurements (§VIII.A, Eqs. 77–88).
  • For a rank-1 PVM {P_k}, the states ρ_k = P_k are orthogonal, so their supports Ω_k are disjoint (Eq. 79).
  • Tr(ρ_k P_k′) = δ_kk′ forces ξ_k = 1 on Ω_k and 0 on the other supports (Eq. 82).
  • Σ_k (1/d) ρ_k = I/d, and every ρ lies in some decomposition of I/d. So preparation noncontextuality gives Σ_k (1/d) µ_k(λ) = µ_{I/d}(λ) (Eq. 88), and ∪_j Ω_j = Ω (Eq. 84).
  • Hence the ξ_k are idempotent everywhere. Coarse-graining extends this to every PVM.
  • “The old proofs are vindicated insofar as they remain proofs of the impossibility of universal noncontextuality”.
• Measurement noncontextuality alone is consistent with quantum theory (§VIII.B, Eqs. 89–92). The Beltrametti–Bugajski model takes Ω to be the rays of Hilbert space, µ_ψ′(ψ) = δ(ψ − ψ′) and ξ_{Q_k}(ψ) = Tr(Q_k|ψ⟩⟨ψ|). It depends only on the POVM, so it is measurement noncontextual. It is not outcome deterministic for sharp measurements, and it is preparation contextual.
• Conclusion (§IX). “One can confine all the contextuality into the preparations and transformations if one likes … On the other hand, one cannot confine all the contextuality into the measurements”. The models Bell and Kochen–Specker gave for 2d “are deemed contextual” under the new definition.

Claims

id	claim	strength	support
C1	The traditional notion is measurement noncontextuality for sharp measurements plus outcome determinism	strong	definitional argument, §III
C2	No preparation-noncontextual ontological model of a qubit exists	strong	proof, §IV, Eqs. 13–36
C3	No transformation-noncontextual ontological model of a qubit exists	strong	proof, §VI and Appendix A
C4	No qubit model is noncontextual for unsharp measurements and outcome-deterministic for sharp ones	strong	proofs, §V.A and §V.B
C5	Preparation noncontextuality over all quantum states implies outcome determinism for sharp measurements	strong	proof, §VIII.A, Eqs. 77–88
C6	Measurement noncontextuality alone is consistent with quantum theory	strong	the Beltrametti–Bugajski model, §VIII.B
C7	Noncontextuality for preparations, transformations and unsharp measurements is as well motivated as for sharp ones	moderate	argument from ontological economy, §VII
C8	Every classical theory is noncontextual for all procedures	weak	asserted in footnote 6, not proved; an opposing view is cited
C9	A failure of local causality in a separable model implies measurement contextuality; the Beltrametti–Bugajski model is nonlocal by nonseparability instead	moderate	informal argument, §IX

Concepts

• Operational equivalence. Two procedures are equivalent when they give the same statistics with every procedure of the other kinds (Eqs. 1–3).
• Context. “Those that are not specified by specifying the equivalence class”. Polaroid versus calcite for a polarisation measurement is the paper’s example. By definition, knowing the context does not improve prediction.
• Ontological model. Ontic states λ in Ω with the representations of Eq. 4. It may have no hidden variables. The Beltrametti–Bugajski model’s ontic states are the quantum states.
• Indicator function. ξ_{M,k}(λ), the probability of outcome k given λ. It is called idempotent, written χ, when it takes only the values 0 and 1.
• Outcome determinism. A measurement represented by idempotent indicator functions. It is distinguished from determinism of the underlying dynamics: the weighted-dice example (§III) is deterministic classical mechanics without outcome determinism.
• Universal noncontextuality. Noncontextuality for preparations, transformations and measurements together.
• Ontic and epistemic views of a set S of states (Eqs. 67–68). In the ontic view, distinct states have disjoint distributions. In the epistemic view, only orthogonal ones do. An ontic view of pure states makes preparation noncontextuality fail trivially (Eqs. 69–76).

Connections

The paper generalises the Bell–Kochen–Specker notion (Bell 1966; Kochen and Specker 1967, both cited, not held). It sets itself against Cabello’s 2003 Kochen–Specker theorem for a single qubit using POVMs. That theorem assumes outcome determinism for unsharp measurements, which the {I/2, I/2} argument shows is inconsistent with measurement noncontextuality anyway. Its notion of measurement noncontextuality follows Bell’s move from local determinism to local causality (footnote 5). Its motivation is “ontological economy”, with Einstein’s equivalence principle as the example (§VII). It does not name Leibniz. The later programme does: Schmid, Selby and Spekkens (LIT-945) recast the definition process-theoretically and derive it from a principle of Leibnizianity. The positive-quasiprobability characterisation in LIT-003, the epistemically restricted theories of LIT-007 and the interference argument of LIT-019 all use the definition given here. Budroni et al. (LIT-263, §IV.E) restate §IV.

Bearing on the record

• THEORY-011: supported, and its promote_when is met for the Spekkens half. §IV’s case analysis is complete and correct. The proof needs only Features 1 and 2 and preparation noncontextuality, as the THEORY says. The §III dimension argument is stated, not proved, but it is elementary: a rank-1 projector has no non-trivial fine-graining. Two corrections:
  • The title’s “no Kochen–Specker argument exists in dimension two” is true of the traditional notion, which uses sharp measurements. Cabello’s 2003 POVM argument for a single qubit is a Kochen–Specker-style proof. Spekkens rejects it, because it assumes outcome determinism for unsharp measurements, not because it fails as mathematics. The THEORY should say “for sharp measurements”.
  • The THEORY’s “What this does not say” ties the choice of notion to “Spekkens’s Leibnizian methodological principle”. This paper argues from ontological economy and Einstein’s equivalence principle, and does not mention Leibniz. The Leibnizian framing is in LIT-945. The substance is the same; the citation for the wording is not this paper.
• THEORY-015: supported, with its own caveat confirmed. The THEORY’s account of §III, §VIII.A–B and §IX is accurate, and the quotation from §IX is exact. The proof does need the full preparation set: Ω is defined through all density operators (Eq. 85), and Eq. 86 holds only because every ρ appears in a decomposition of I/d. In a fragment with fewer preparations, the step from Eq. 83 to idempotence fails as written. Two small precisions:
  • The paper extends from rank-1 PVMs to all PVMs by coarse-graining. That is a statement about measurements implemented by coarse-graining. The same support argument, run directly on ρ_k = P_k/rank(P_k), covers the rest. So the THEORY’s “every projective measurement” stands.
  • What the old proofs refute is preparation plus measurement noncontextuality. Universal noncontextuality includes both, so the THEORY’s wording is right. A reader should not take it to mean that they refute preparation noncontextuality alone. That needs §IV. The join to the sheaf form (LIT-016) remains the record’s own, as THEORY-015 says.
• The seeding checks.
  • The first, §VIII.A’s reliance on the full preparation set, is confirmed above.
  • The second is confirmed in the paper’s own words: §V’s proofs are “contingent on” outcome determinism for sharp measurements. §VIII says that neither Bell’s, nor Kochen and Specker’s, “nor any of the proofs of these types including those presented in section V” rule out measurement noncontextuality. They are proofs against preparation plus measurement noncontextuality.
  • The third is confirmed: “By our definition, any classical theory is necessarily noncontextual for all experimental procedures” is a footnote assertion with no proof, and it cites an opposing view (Bacciagaluppi).
• QUESTION-067. The paper bears on the question in three ways.
  • It supports the question’s premise. Preparation contextuality shows in a scenario with no overlapping measurement contexts. §IV uses six preparations and needs nothing from measurements beyond single-shot distinguishability. A sheaf test cannot see it.
  • It sharpens the condition. Operational equivalence is defined over all measurements (Eq. 1). An equivalence over a restricted probe family, such as the one CLAIM-142 and CLAIM-046 define, is Spekkens’s relation only if that family is tomographically complete for the renderings. Otherwise two renderings the probes cannot separate may still be operationally inequivalent, and the assumption does not apply.
  • It names the obstacle a pragmatic test would meet. Finite precision “might imply that in practice no two experimental procedures are found to be operationally equivalent”. The paper’s proposed fix is that operationally similar procedures get similar representations. It leaves that without a measure of similarity.
• Feature 1 matters for the mixed-preparation reading. QUESTION-067, after CLAIM-044, reads an ambiguous utterance as a mixed preparation with several decompositions. The §IV proof needs the two members of each two-element decomposition to be perfectly distinguishable (orthogonal states). Readings that are only probably distinguishable do not give Eqs. 24–26. A pragmatic version would need either such pairs, or a noise-robust inequality, which this paper does not supply.
• No instruction for machine-learning practice; nothing for the anthology.

Limitations

• All results are about exact operational equivalence and exact distinguishability. There is no inequality and no noise tolerance. §IX names the finite-precision problem and leaves it open.
• §VIII.A is proved for quantum theory with every density operator available as a preparation. It is not shown for subtheories or for general operational theories.
• That classical theories are always noncontextual (footnote 6) is asserted. The claim that contextuality is therefore nonclassicality rests on it.
• The motivation for noncontextuality (§VII) is an argument from ontological economy and from the “conspiratorial” look of context-dependence that averages out. It is not a derivation, and the paper notes that Bell and Mermin judged the matter differently.
• The promised conversion of a preparation-contextuality proof into a nonlocality proof is deferred to another paper (Rudolph and Spekkens, “in preparation”).
• Transformation contextuality is proved only through its reduction to the §IV structure.

Open questions

• A noise-robust form: what replaces exact equivalence? Answered later in the programme by noncontextuality inequalities, which the record does not hold.
• Does §VIII.A survive in fragments that lack some preparations, such as stabilizer or interferometric fragments (LIT-019)? It would be settled by a proof of outcome determinism from the preparations a given fragment contains, or a fragment with a preparation-noncontextual model that is not outcome deterministic for its sharp measurements.
• Can contextuality be shown to power a quantum computational advantage? §IX speculates that it might.