Testing locality or noncontextuality with lowest moments

Abstract

The quest for fundamental test of quantum mechanics is an ongoing effort. We are addressing the question of what are the lowest possible moments to prove quantum nonlocality and noncontextuality without any further assumption - in particular without the often assumed dichotomy. We first show that second order correlations can always be explained by a classical noncontextual local hidden variable theory. Similar third order correlations cannot violate classical inequalities as well in general, except for a special state-dependent noncontextuality. However, we show that fourth order correlations can violate locality and state-independent noncontextuality. Finally we obtain a fully scalable continuous variable Bell inequality, which might be useful in Bell tests closing all loopholes simultaneously.