Often we hear from young researchers that they have violated a Bell inequality with their new source by this and this factor. In such cases we usually forget about three basic underlying assumptions. Two of them are well known; Realism stating that outcomes of measurements exist before they are revealed in a measuring act and Locality forbidding superluminal communication between spatially separated laboratories. The third important assumption is Freedom exercised by the observers to choose their local measurements independently of each other.
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Michael J. W. Hall from the Australian National University investigates in [http://prl.aps.org/abstract/PRL/v105/i25/e250404 Phys. Rev. Lett. 105 250404] consequences of giving up this freedom - gradually. He considers local hidden variables, the same as in the usual Bell theorem, but distributed dependently on what is measured on both sides. The difference in the hidden variable distribution for different pair of observables quantifies the ability of the observers to be choose these settings independently.

Hall shows that with some measurement choice dependence the CHSH expression can even attain its maximal algebraic value of 4. This is in contrast to 2 being the classical limit, and to 2.82, as quantum mechanics predicts. Thus the question arises, how much freedom should the observers give up to reconstruct the quantum mechanical prediction. About 14 percent thereof turns out to be sufficient. It might be interesting to observe how this result relates to other known fundamental limitations of the general theories of the universe.</p>