  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 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.
