The heralded pair is modelled as a singlet with a fraction (1 − V) dephased into the classical mixture (up, down) + (down, up), with V = 2F − 1 for a state fidelity F. Each lab's readout is asymmetric: a true 0 reads as 0 with probability F₀ and a true ±1 reads as ±1 with probability F₁. The angles are the experiment's own, including the small offset ε = 0.026π. Each cloud is used for the pair (shared phase) and for the dephased part (each electron prepared on its own).
F₀ follows from the average: F₀ = 2 × average − F₁. Moving F₁ changes S by less than 0.01, so only the averages matter.
Observed: S = 2.42 ± 0.20 from 245 trials, with k = 196 correlated outcomes (the paper's convention). Expected k = 245 × (½ + S/8), assuming equal numbers of trials for each setting pair.
| Model | Ideal S at these angles | Expected S | Data minus model (in σ of 0.20) | Expected k | Observed 196 minus expected (in σ) |
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Quantum theory and the exact half-cosine distribution predict the same expected S, because each component of the mixture matches: the shared phase gives −cos Δ and a prepared electron gives cos θ. The von Mises cloud predicts a slightly lower value because its prepared-spin curve only reaches 0.93. The differences between the models (about 0.05 in S at most) are far below the data's error bar of 0.20 and the model's own uncertainty from F. Distinguishing them at one standard deviation would take on the order of 4,000 trials, against 245 in 220 hours.
Assumptions: all of the state's infidelity is treated as loss of coherence (a population error would lower S by about 0.03 for every 0.02), and the readout average is taken to be the paper's "combined initialisation and single-shot readout fidelity". The paper's own state model is in its supplementary information, which was not read. The data cannot test the dependence between the shared phase and the settings.