In a local hidden-variable description of a Bell experiment, shared hidden data have a distribution independent of the measurement settings. Each outcome depends on the local setting and those shared data, without dependence on the remote setting. Local randomness can be included in the hidden data. Perfect correlations then constrain simultaneous assignments to the possible local measurements. The GHZ theorem shows an all-versus-nothing inconsistency between such assignments and certain three-qubit correlations.
If a successful preparation flag is fixed before freely selected Bell inequality settings, conditioning a local hidden-variable theory on that flag only replaces its hidden-variable distribution by the conditional distribution. The later response functions remain local and the distribution is independent of the later settings, so the CHSH inequality still holds. Setting-dependent rejection during the Bell measurements does not satisfy this argument.
Outcome independence requires . It says that, once settings and the proposed complete state are fixed, learning the distant outcome does not change the local outcome probability. Combining it with parameter independence gives Bell factorization. Entangled quantum states generally fail outcome independence even though their observable marginal distributions obey quantum no-signalling.
For spacelike-separated measurement regions, a sufficient specification of their relevant past should screen each outcome from additional information in the other wing. In a two-setting experiment a Bell-local model has . Together with measurement independence and bounded outcomes, this implies the CHSH inequality. Bell local causality is stronger than quantum no-signalling and is not an assumption that every outcome was predetermined.
Outcome determinism says that a complete hidden state and the measurement settings fix each measurement outcome with probability one.
Parameter independence says that, conditional on the complete hidden state, the probability distribution of one party's outcome is independent of the other party's measurement setting.
Measurement independence says that the distribution of the hidden state is independent of the measurement settings: . Its failure is the measurement-dependence loophole in a Bell test.
A Bell-test loophole in which hidden information determining the device outputs is correlated with the chosen measurement settings. The usual local CHSH inequality averages a common hidden-variable distribution over independently selected settings. If that distribution can instead depend on the selected pair, the four counterfactual assignments need not obey the same averaging argument. An adversary controlling both the devices and their setting generators can exploit this dependence without violating the local-bound proof's actual hypotheses.
