Testing quantum correlations versus single-particle properties within Leggett's model and beyond

Quantum theory predicts and experiments confirm that nature can produce correlations between distant events that are nonlocal in the sense of violating a Bell inequality. Nevertheless, Bell's strong sentence {\it Correlations cry out for explanations} remains relevant. The maturing of quantum information science and the discovery of the power of nonlocal correlations, e.g. for cryptographic key distribution beyond the standard Quantum Key Distribution schemes, strengthen Bell's wish and make it even more timely. In 2003, Leggett proposed an alternative model for nonlocal correlations [Found. Phys. 33, 1469 (2003)], that he proved to be incompatible with quantum predictions. We present here a new approach to this model, along with new inequalities for testing it. Remarkably these inequalities can be derived in a very simple way, assuming only the non-negativity of probability distributions; they are also stronger than previously published Leggett-type inequalities. The simplest of these inequalities is experimentally violated. Then we go beyond Leggett's model, and show that one cannot ascribe even partially defined individual properties to the components of a maximally entangled pair.