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.

Figure 1

Scenario of bilocality, the network we consider in this work. In the quantum setting, two independent sources distribute entangled states, ${\rho }_{AB}$ and ${\rho }_{BC}$, between three distant observers—Alice, Bob, and Charlie. In order to compare the resulting quantum correlations to classical ones, we discuss 2-local correlations obtained by two independent sources of shared classical random variables, $\lambda$ and $\mu$. For the bilocality inequality we consider, Alice and Charlie perform two dichotomic measurements, while Bob performs a fixed measurement with four possible outcomes. In the quantum setting, Bob's measurement is taken to be the Bell state measurement.