All entangled pure quantum states violate the bilocality inequality

The nature of quantum correlations in networks featuring independent sources of entanglement remains poorly understood. Here, focusing on the simplest network of entanglement swapping, we start a systematic characterization of the set of quantum states leading to violation of the so-called "bilocality" inequality. First, we show that all possible pairs of entangled pure states can violate the inequality. Next, we derive a general criterion for violation for arbitrary pairs of mixed two-qubit states. Notably, this reveals a strong connection between the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality and the bilocality inequality, namely, that any entangled state violating CHSH also violates the bilocality inequality. We conclude with a list of open questions.