Exact binomial alternatives

For the local correlation measurements, orient their outcomes as $X\otimes X$, $-Y\otimes Y$, and $Z\otimes Z$, and let $q_i$ be the respective wrong-outcome probabilities. Then

Allocating ${\delta }_i={\alpha }_{\mathrm{s}}/3$ and using the three one-sided endpoints gives

When no wrong outcome is seen and all settings have $n$ shots, passing threshold $\epsilon$ requires

If a joint Bell-projector measurement is independently available and calibrated, it directly measures $f$. With no failures in $n$ shots, its one-sided lower endpoint is $f_L={{\alpha }_{\mathrm{s}}}^{1/n}$. The corresponding requirement is

This uses a different measurement architecture, which must be listed in the validation contract.

For $\epsilon =0.1$, ${\alpha }_{\mathrm{s}}=0.01$, and zero calibration and transfer allowance:

These counts illustrate the procedure dependence of the forty-shot result. Nonzero preparation, measurement, transfer, or physical error changes the required counts.