Non-Contextuality

Actualisation writes a definite record (Axiom R). The record answers a specific question: “what is the outcome of this measurement on this state?”

The next question is: can the probability of the outcome depend on what other measurements are performed simultaneously?

In quantum mechanics, two observables A and B are compatible (they can be measured simultaneously) if and only if they commute: [A, B] = 0. Contextuality would mean that the probability of getting outcome a when measuring A depends on whether A was measured alone, or alongside compatible observable B, or alongside compatible observable C.

The same measurement, the same state, but different probabilities depending on context.

Proposition 2 (Non-Contextuality). Let |ψ⟩ be a pure state and P_a a projector corresponding to measurement outcome a. The probability μ(P_a) assigned by actualisation depends only on |ψ⟩ and P_a.

It does not depend on which resolution of the identity P_a belongs to. Actualisation is Gleason non-contextual.

Proof. Axiom B provides one break. Axiom R writes one record for that break. The record registers the outcome of the actualisation event.

Identify what determines the record by examining what the axioms provide to the actualisation process.

Input 1: the state |ψ⟩ (the pre-state being actualised). Input 2: the projector P_a (the outcome registered by the break). These two inputs are the complete specification of the actualisation event under the axioms.

Axiom S provides the distinction (the outcome is distinguished from alternatives). Axiom B provides the single break. Axiom R writes the record for the state and the outcome.

Now suppose the probability of outcome a depended on context — on which resolution of the identity {P_1, …, P_n} the projector P_a belongs to.

This context would constitute a third input to the actualisation process: the record would depend on (|ψ⟩, P_a, {P_1, …, P_n}). But the axioms provide no mechanism for this third input.

Axiom B says: one break. The break is the actualisation of the state by the projector. It does not include a specification of what other measurements could have been performed.

The break has no “compatible observable” register. Axiom R says: the record registers what happened — outcome a — not what could have happened.

The counterfactual alternatives (the other projectors in the resolution) are properties of the measurement apparatus on the manifold. They determine which outcomes are possible but are not inscribed in the record.

The record contains the state and the outcome. Context is not recorded because context is not an event.

Therefore: μ(P_a) = f(|ψ⟩, P_a) for some function f that depends only on the state and the projector.

Equivalently: the probability assigned to P_a does not depend on which resolution of the identity P_a belongs to. Actualisation is Gleason non-contextual.

The exhaustiveness of the two-input specification is the completeness of {S, B, R, C} doing its work — the obligation fenced at KS-P.1. A third input to actualisation would be structure beyond the four axioms: a fifth axiom. Contextual hidden-variable proposals are therefore fifth-axiom proposals, and they are answered where fifth axioms are answered — at the completeness result and its fence. ∎