What Bell Does and Does Not Prove

Bell’s theorem shows that some quantum correlations cannot be reproduced by a setting-independent Bell-local model. An experiment can reject that model class under its implemented protocol and statistical assumptions. Neither statement alone identifies a unique interpretation, prohibits all hidden variables or permits faster-than-light communication.

The conjunction actually tested

The Bell theorem states the factorized model and derives its consequence. Its essential structure is

$$p(a,b\mid x,y)=\int\mu(d\lambda)\, p_A(a\mid x,\lambda)p_B(b\mid y,\lambda),$$

with a setting-independent measure $\mu$. A violation means there is no representation of this form for the tested distribution. Terms such as “local realism” are useful only when unpacked into these mathematical premises and their physical interpretation.

Local determinism is not an extra universal assumption needed to derive CHSH: stochastic local response functions obey the same bound. In a finite scenario they can be represented as mixtures of deterministic strategies. Abandoning determinism alone therefore does not restore a Bell-local account of the violating data.

Five common overstatements

“All hidden variables are impossible.” Nonlocal hidden-variable models are not excluded by a theorem about local factorization. Pilot-wave theories illustrate why ontology and locality are separate questions.

“Realism has been disproved.” Realism is not one precise mathematical condition. A claim about actual configurations, definite outcomes, observer-independent laws or an ontic quantum state must be assessed separately.

“Entanglement is exactly Bell nonlocality.” Entanglement belongs to a state and subsystem split; Bell nonlocality belongs to achievable input–output correlations. Mixed states and restricted measurement choices distinguish the notions.

“A violation is a superluminal communication channel.” Quantum local marginals remain independent of remote settings. No-Signaling proves the relevant statement and distinguishes conditioning from communication.

“A nonviolation proves locality.” A failed witness can result from unsuitable measurements, noise or insufficient statistical power. A general local model requires more than one nonviolating number.

What changing a premise requires

A proposal that relaxes measurement independence must specify the setting–hidden-variable relation and show how it recovers the observed statistics without losing predictive content. A retrocausal account must state the causal structure and boundary conditions it uses. A nonlocal theory must explain how its nonlocal dynamics remains consistent with observed signaling constraints.

These are possible changes of theoretical premise, not experimental loopholes that ordinary detector improvements can all eliminate. They should be evaluated by internal consistency, explanatory specificity and empirical adequacy. Superdeterministic Approaches and Retrocausal Approaches examine the relevant proposals without treating their labels as complete models.

Exercise: locate an unjustified conclusion

An experiment rules out a declared local null with p-value below $10^{-6}$. A report says, “There is less than a one-in-a-million chance that any hidden-variable theory is true.” Give two independent problems with the statement.

Solution

The null covers the specified local model class, not every hidden-variable theory. Also, a p-value is a bound on data extremeness under that null and protocol, not the posterior probability of a theory. Obtaining posterior model probabilities would require prior probabilities and likelihoods for explicit alternatives.

References

• Bell, John S., “On the Einstein Podolsky Rosen paradox,” Physics Physique Fizika 1, 195–200 (1964), doi:10.1103/PhysicsPhysiqueFizika.1.195.
• Bell, John S., Speakable and Unspeakable in Quantum Mechanics, 2nd edition, Cambridge University Press (2004) — local causality and interpretation.
• Brunner, Nicolas, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014), doi:10.1103/RevModPhys.86.419.