Source
Pusey, Barrett & Rudolph (2011), LIT-062, read in full in NOTE-030.
What was actually shown
Assumptions. A system prepared in isolation in a pure state has a physical state λ, distributed as µ_ψ (p. 1; pure states only). Independent preparations of n systems give the product µ_{x1}(λ1)⋯µ_{xn}(λn), and the joint physical state is exactly the tuple (λ1, …, λn) (Eqs. 4–5, B3). Measurement outcomes depend only on that tuple, through nonnegative response functions (Eq. 5, B1). Determinism is not assumed (p. 4), nor Bell locality for the measurement.
Simple case (pp. 2–3, Eq. 1). For |0⟩ and |+⟩ and two copies, an entangled basis has each element orthogonal to one of |00⟩, |0+⟩, |+0⟩, |++⟩. Any overlap of µ_0 and µ_+ then gives a quantum-forbidden outcome a nonzero rate.
General case (App. A). For |ψ0,1⟩ = cos(θ/2)|0⟩ ± sin(θ/2)|1⟩ and n with 2 arctan(2^{1/n} − 1) ≤ θ, a circuit H^{⊗n}R_αZ_β^{⊗n} followed by a computational-basis measurement gives each outcome |x⃗⟩ zero probability on the preparation |Ψ(x⃗)⟩. NOTE-030 computes the minimum angle falling with n: 90°, 45°, 29.1°, 21.4°, 16.9° for n = 1…5.
Noise-robust theorem (Eq. 7 = B2). If every outcome probability is within ε of quantum theory, then D(µ0, µ1) = ½∫|µ0 − µ1|dλ ≥ 1 − 2ε^{1/n}; equivalently ω(µ0, µ1)ⁿ ≤ 2ⁿε (B11). The proof: the overlap of the product distributions factorises as ωⁿ (B7), while the forbidden outcomes bound it by 2ⁿε (B10). At ε = 0 the supports meet in measure zero.
What could have come out otherwise. Before this, whether any model of this kind could let distinct pure states share ontic states was open (p. 4). A ψ-epistemic model satisfying the assumptions could have existed. It cannot.
What this does not say
- That the quantum state is real, without qualification. The theorem is conditional on preparation independence in its strong form. The authors concede that dropping it permits ψ-epistemic models (Lewis et al., their ref. [16]) and answer only by calling such models “contrived” (p. 4; NOTE-030, C4–C5).
- That epistemic views are refuted. The abstract’s informal wording, that ψ cannot represent “mere information”, is broader than the theorem, which defines mere information as support overlap (p. 2; C6). The paper leaves the QBist option and theories outside ontological models open (p. 5). What the theorem excludes is THEORY-586’s subject.
- Anything about mixed states. Only pure states are treated.
- That ψ-epistemic models of quantum fragments are excluded. Spekkens’ epistricted theories are ψ-epistemic with product composition and reproduce stabilizer and Gaussian subtheories exactly. By PBR’s own argument those subtheories contain no PBR-type measurement, and PBR’s measurements are indeed not stabilizer operations (NOTE-030’s check). This consequence is NOTE-030’s inference; none of the papers states it.
- That the measurement uses the fewest systems possible. Minimality in n rests on numerical SDP solutions only (App. C).
- That the experiment has been done. Feasibility is asserted (p. 4).
Connections
- LIT-003 (Schmid, Selby, Pusey, Spekkens) derives PBR’s product form from diagram preservation (its Prop. 4.6), so a diagram-preserving ontological model of full quantum theory is caught by PBR (NOTE-030).
- Gao’s protective-measurement route to ψ-ontology, claimed to avoid preparation independence, is THEORY-630.
- Rovelli’s relational reading sidesteps the theorem by positing no ontic state (THEORY-621). Cuffaro & Hartmann place orthodox readings outside the framework (THEORY-631).
- Wave-function realism’s consequences for causal closure (THEORY-485) read PBR the same way.
- The toy bit theory (THEORY-474) and the interference model (THEORY-403) are ψ-epistemic models of fragments, outside the theorem’s reach.