Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 2 / f / Solution

No. The independent-setting assumption can fail if Eve controls the random-number generators. Her hidden information can then include the settings that will be selected, and the outcome assignment can depend on that information. For each known setting pair she can prearrange its one tested winning relation, without having to satisfy all four counterfactual relations at once.

For example, arrange equal outputs for the first three setting pairs and opposite outputs for the fourth, conditioning the programmed outputs on the known pair. This can give success one with predetermined records. It does not contradict the $3/4$ proof, which assumes the same hidden-assignment distribution for every independently selected pair. This is the measurement-independence loophole, so the claimed security conclusion is not established by such a compromised test.