Bell Assumptions Checklist

A Bell test compares observed correlations with a local model under stated assumptions. This checklist separates the mathematical premises from the experimental conditions needed to apply them. The derivation belongs to Bell Theorem and CHSH Inequality.

Assumptions of a Bell-local model

For settings $x,y$, outcomes $a,b$, and hidden variable $\lambda$, check the following.

Factorization. Does the model assume $p(a,b\mid x,y,\lambda)=p(a\mid x,\lambda)p(b\mid y,\lambda)$? The variable $\lambda$ is intended to contain the relevant shared prior information. A failure to screen off correlations after conditioning on an incomplete variable is not yet a Bell violation.

Measurement independence. Is $\mu(\lambda\mid x,y)=\mu(\lambda)$ assumed? State how input choices are generated and what causal independence is required. “Free choice” is shorthand for a probabilistic premise, not an experimental resolution of philosophical free will.

Outcome and parameter independence. If used separately, define them precisely. Their conjunction yields factorization. With a measurement-independent hidden-variable distribution, parameter independence conditioned on $\lambda$ implies observable no-signaling. Without that distributional assumption, the implication need not hold.

Stochastic responses. Determinism need not be added as an independent premise. The local class already permits stochastic response functions; extra local randomness can be absorbed into a larger hidden-variable description.

Conditions for applying the bound to data

Check	What must be specified	Failure mode
Trial definition	When a trial starts and which inputs and outputs belong together	Coincidence selection can depend on settings or outcomes
Detection	Treatment of losses, no-click events, and rejected data	Fair-sampling assumptions silently enter postselection
Locality	Relevant choice and outcome events and their spacetime separation	Sub-luminal communication can explain correlations
Input independence	Randomness source and permitted correlations	Devices may anticipate settings
Statistical analysis	Test statistic, stopping rule, confidence statement	A nominal violation can overstate evidence
Memory	Permitted dependence between trials	An independent-trial calculation is used outside its assumptions
Heralding	Whether successful trial selection occurs before settings	Later selection can correlate accepted inputs and hidden variables

Bell Loopholes explains the physical mechanisms behind these checks.

A postselection diagnostic

Suppose detection probability depends on both the setting and $\lambda$. Among detected events,

$$\mu(\lambda\mid x,y,\mathrm{detected}) \propto p(\mathrm{detected}\mid x,y,\lambda)\mu(\lambda).$$

The effective hidden-variable distribution can then depend on settings even when the original source distribution did not. Applying the unmodified local bound to the selected sample requires justification.

The issue is not that conditioning is mathematically forbidden. It is that conditioning can change the model class for which the bound was proved.

Reporting the conclusion

State which local model is excluded, by which statistic, at what confidence, and under which experimental assumptions. A Bell violation does not certify every device property or automatically produce a secure key. Device-Independent Reasoning explains the additional inferential steps.

Avoid replacing the conclusion with “realism is false” or “signals traveled faster than light.” Both omit distinctions essential to the theorem.

References

• Brunner, Nicolas, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014). Review.
• Larsson, Jan-Åke, “Loopholes in Bell inequality tests of local realism,” Journal of Physics A: Mathematical and Theoretical 47, 424003 (2014). Review.