Measurement independence

Measurement independence says that the distribution of the hidden state is independent of the measurement settings: $\rho (\lambda |a,b)=\rho (\lambda )$. 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.