Given a sequence of pairs of spin-one half particles in the singlet state,assume that Alice measures the normalized projections along some vector of thespins of one vector per pair along that vector while Bob measures thenormalized projections along some vector of the spins of the other member ofeach pair. Then Quantum Mechanics, or QM, lets one evaluate the correlation ofthe projections along these two vectors as minus the cosinus of the anglebetween said vectors; we assume that all vectors are chosen in a fixed plane.Assuming Classical Microscopic Realism, or CMR, there exist also normalizedprojection pairs of the spins of the pairs of particles along some other pairof vectors. Assuming QM and MR, we also have that the correlations of theprojections along the other vectors as minus the cosinus of the angle betweenthe extra vectors. Assuming Locality,i.e., the impossibility of any effect ofan event on another event when said events are spatially separated, beside QMand MR, the theory of Bell lets one deduce various violations of someinequalities at some choices of quadruplets of the vectors that have beenchosen. Our main result is the existence of quadruplets where at least one ofthe said inequalities is violated if one only assumes QM, MR and some very mildfurther hypotheses. These weak hypotheses only concern the behavior ofcorrelations that we use near special quadruplets. We thus get versions ofBell's theorem that are strictly stronger than the original one and inparticular do not assume Locality.
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