The aim of this chapter is to attract attention of experimenters to the originalBell (OB) inequality which was shadowed by the common considerationof the CHSH inequality. Since this chapter is directed to experimenters, herewe present the standard viewpoint on the violation of the Bell inequality andthe EPR argument. There are two reasonsto test the OB inequality and not the CHSH inequality. First, theOB inequality is a straightforward consequence of the EPR argumentation.And only this inequality is related to the EPR–Bohr debate.The second distinguishing feature of the OB inequality was emphasizedby Pitowsky. He pointed out that the OB inequality provides a higherdegree of violations of classicality than the CHSH inequality. Thus, by violating the OBinequality it is possible to approach a higher degree of deviation from classicality.The main problem is that the OB inequality is derived under theassumption of perfect (anti-)correlations. However, the last few years have been characterizedby the amazing development of quantum technologies. Nowadays,there exist sources producing with very high probability the pairs of photonsin the singlet state. Moreover, the efficiency of photon detectors wasimproved tremendously. In any event one can start by proceeding with thefair sampling assumption.

In this chapter we introduce the general class of symmetric two-qubit statesguaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomeswhenever some spin observable is measured at both sites. It is proventhat, for all states from this class, the maximal violation of the original Bellinequality (OB) is upper bounded by 3/2 and specify the two-qubit stateswhere this quantum upper bound is attained. The case of two-qutrit statesis more complicated. Here, for all two-qutrit states, we obtain the same upperbound 3/2 for violation of the original Bell inequality under Alice and Bobspin measurements. But it has not yet been shown that this quantum upperbound is the least one. The experimental consequences of this mathematicalstudy are discussed.