A herald defines which attempt becomes a trial
Hensen and colleagues used a central photonic herald to prepare distant electron spins for a Bell test in 2015. Successful preparation was identified before the local measurement choices. The subsequent efficient spin readout supplied an outcome on each accepted trial.
Compare two records. In the first, a preparation flag is fixed before fresh inputs are generated; every ensuing output is retained. In the second, inputs and outputs already exist and only attractive coincidences are kept. Both select data, but their causal ordering creates very different possibilities for a local model.
Read the evidence: B. Hensen et al. (2015), Experimental Loophole-Free Violation of a Bell Inequality Using Entangled Electron Spins Separated by 1.3 km — author manuscript, arXiv:1508.05949
A small game makes the local bound visible
Use binary input bits and output bits for an explanatory CHSH game. The winning rule below asks the output parity to equal the product of the inputs. Each side may know a shared instruction but must answer using only its own input. Inputs are independent of that instruction and uniformly distributed.
There are four input pairs. A deterministic local instruction assigns two possible answers on each side. The parity requirements for the first three pairs force an answer that loses the fourth, or another equivalent single failure. Shared random mixtures of instructions cannot improve the maximum average beyond three wins in four.
Read the evidence: D. Elkouss and S. Wehner (2016), (Nearly) Optimal P-Values for All Bell Inequalities — author manuscript, arXiv:1510.07233v3; npj Quantum Information 2, 16026
Keep the reported statistic and the score separate
The 2015 spin experiment recorded 245 trials and 196 game wins. It also reported a CHSH correlation value of 2.42 with a conventional standard deviation of 0.20. Setting frequencies were not exactly equal, so the correlation estimator and an overall win-frequency conversion need not coincide numerically.
For a quick bookkeeping check, the win fraction is four fifths. The conversion that assumes exactly equal input frequencies gives 2.4, not 2.42. This difference is not evidence that the report is inconsistent; it shows why one must preserve the statistic’s definition instead of replacing it with a convenient rounded proxy.
Read the evidence: B. Hensen et al. (2015), Experimental Loophole-Free Violation of a Bell Inequality Using Entangled Electron Spins Separated by 1.3 km — author manuscript, arXiv:1508.05949D. Elkouss and S. Wehner (2016), (Nearly) Optimal P-Values for All Bell Inequalities — author manuscript, arXiv:1510.07233v3; npj Quantum Information 2, 16026
Device memory need not erase the bound
If inputs remain fresh and independent on every trial, a local device may use all earlier inputs and outcomes and still face the same conditional upper bound on winning. A finite-sample analysis can use that per-trial constraint without assuming identically distributed apparatus behaviour.
For a fixed number of trials and ideal uniform inputs, the upper tail of a binomial distribution with success probability three quarters provides the game-score bound. Predictable inputs can change that bound; choosing when to stop can change the applicable test. The actual experiment’s treatment of imperfections must therefore accompany its reported p-value.
Read the evidence: D. Elkouss and S. Wehner (2016), (Nearly) Optimal P-Values for All Bell Inequalities — author manuscript, arXiv:1510.07233v3; npj Quantum Information 2, 16026
A p-value is conditional evidence
Hensen’s analysis allowing device memory and measured setting imperfections reported a p-value of approximately 0.039. That is a statement about the extremeness of a statistic under a specified local null model. It is not a 3.9 percent posterior probability that locality is true.
For an audit, demand the accepted-trial rule, scoring convention, setting assumptions and stopping rule before quoting a significance number. A normal-error summary and a test allowing adversarial memory answer different statistical questions. Neither makes an independent setting assumption unnecessary.
Read the evidence: B. Hensen et al. (2015), Experimental Loophole-Free Violation of a Bell Inequality Using Entangled Electron Spins Separated by 1.3 km — author manuscript, arXiv:1508.05949D. Elkouss and S. Wehner (2016), (Nearly) Optimal P-Values for All Bell Inequalities — author manuscript, arXiv:1510.07233v3; npj Quantum Information 2, 16026
Different platforms solve the loss problem differently
Giustina and colleagues’ 2015 photon experiment used a different inequality and trial treatment with high-efficiency detection. It should not be rewritten as the spin experiment with photons substituted. Event-ready spin trials and optical trials with explicit nondetections require their own score and timing contracts.
The linked Labs explain sampling and conditional quantum protocols in ideal models. They do not reproduce the 245 original trials, detector calibration or spacetime certification. The conclusion concerns local models satisfying the stated causal and input assumptions; it is neither a messaging protocol nor an elimination of every imaginable hidden-variable theory.
Read the evidence: Giustina et al. (2015), Significant-loophole-free test of Bell’s theorem with entangled photons