  In this paper the failure of Hardy's nonlocality proof for the class of
maximally entangled states is considered. A detailed analysis shows that the
incompatibility of the Hardy equations for this class of states physically
originates from the fact that the existence of quantum perfect correlations for
the three pairs of two-valued observables (D_11,D_21), (D_11,D_22) and
(D_12,D_21) [in the sense of having with certainty equal (different) readings
for a joint measurement of any one of the pairs (D_11,D_21), (D_11,D_22), and
(D_12,D_21)], necessarily entails perfect correlation for the pair of
observables (D_12,D_22) [in the sense of having with certainty equal
(different) readings for a joint measurement of the pair (D_12,D_22)]. Indeed,
the set of these four perfect correlations is found to satisfy the CHSH
inequality, and then no violations of local realism will arise for the
maximally entangled state as far as the four observables D_ij, i,j = 1,2, are
concerned. The connection between this fact and the impossibility for the
quantum mechanical predictions to give the maximum possible theoretical
violation of the CHSH inequality is pointed out. Moreover, it is generally
proved that the fulfillment of all the Hardy nonlocality conditions necessarily
entails a violation of the resulting CHSH inequality. The largest violation of
this latter inequality is determined.
