1211.6407 (S. Aravinda et al.)

On a general criterion for nonclassicality    [PDF]

S. Aravinda, R. Srikanth

Intuitively, the quantum violation of the Leggett-Garg (LG) inequality is less surprising than that of Bell-type inequalities or contextuality inequalities because the latter two involve correlations between commuting observables, and hence do not lead to signaling, whereas the former involves `signaling in time'/memory. We derive a general Bell-type inequality, encompassing both the spatial and temporal variants, where the classical bound is determined by the maximum violation of the inequality allowed by the signaling in the correlations. This is used to make precise the notion of non-classicality, and in particular, to explain the sense in which some quantum violations of the LG inequality are classical, though sufficiently large quantum violations will be non-classical. Further, the violation of the CHSH inequality is shown to be in general more non-classical than that of its temporal variant for the same level of violation. Our approach demonstrates the nonclassicality of a qubit, for which proofs based on non-contextuality and Bell-type inequalities, which apply only in dimensions greater than 2, do not exist. We find that quantum invasiveness is, apart from nonlocality and contextuality, a fundamental nonclassical resource in QM, related ultimately to non-commutativity. Quite generally, a theory may be called non-classical if it entails effects that cannot be explained by the signaling available within the theory. Finally, we draw attention to an analogy between non-classicality and the meta-mathematical concept of (G\"odel) incompleteness.

View original: http://arxiv.org/abs/1211.6407