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Quantum Bell inequalities from Information Causality -- tight for Macroscopic Locality

In a Bell test, the set of observed probability distributions complying with the principle of local realism is fully characterized by Bell inequalities. Quantum theory allows for a violation of these inequalities, which is famously regarded as Bell nonlocality. However, finding the maximal degree of this violation is, in general, an undecidable problem. Consequently, no algorithm can be used to derive quantum analogs of Bell inequalities, which would characterize the set of probability distributions allowed by quantum theory. Here we present a family of inequalities, which approximate the set of quantum correlations in Bell scenarios where the number of settings or outcomes can be arbitrary. We derive these inequalities from the principle of Information Causality, and thus, we do not assume the formalism of quantum mechanics. Moreover, we identify a subspace in the correlation space for which the derived inequalities give the necessary and sufficient conditions for the principle of Macroscopic Locality. As a result, we show that in this subspace, the principle of Information Causality is strictly stronger than the principle of Macroscopic Locality.