Escaping the Shadow of Bell's Theorem in Network Nonlocality

The possibility of nonclassicality in networks unrelated to Bell’s original eponymous theorem has recently attracted significant interest. Here, we identify a sufficient condition for being “outside the shadow of Bell’s theorem” and introduce a testable criterion capable of certifying the novelty of instances of network-nonclassicality which we call minimal network nonclassicality. We provide examples of minimally network nonclassical correlations realizable in quantum theory as well as examples coming from more exotic operational probabilistic theories. In particular, we apply these concepts to the simplest configuration of the 3-chain scenario (a.k.a. the bilocality scenario) to prove that certain correlations have escaped the shadow of Bell's theorem. While some of the examples herein are unprecedented, we also revisit more familiar examples of network nonclassicality in order to highlight the contrast between our approach versus prior approaches with respect to assessing novelty.