Self-testing nonlocality without entanglement

Quantum theory allows for nonlocality without entanglement. Notably, there exist bipartite quantum measurements consisting of only product eigenstates, yet they cannot be implemented via local quantum operations and classical communication. In the present work, we show that a measurement exhibiting nonlocality without entanglement can be certified in a device-independent manner. Specifically, we consider a simple quantum network and construct a self-testing procedure that is robust to noise. This result also demonstrates that genuine network quantum nonlocality can be obtained using only nonentangled measurements. From a more general perspective, our work establishes a connection between the effect of nonlocality without entanglement and the area of Bell nonlocality.

Figure 1

We consider the entanglement swapping (or bilocality) network represented in (a) where the middle party Bob performs measurements on two incoming subsystems. Our goal is to self-test one of these measurements, denoted ${\mathsf{M}}_{◊}^{\prime }$, which features nonlocality without entanglement. That is, the measurement consists of only product eigenstates [as shown by the domino tiles in (b)] yet it cannot be implemented by LOCC. The red tile corresponds to the two projectors ${\mathsf{M}}_{0,1|◊}^{\prime }$ and ${\mathsf{M}}_{0,2|◊}^{\prime }$ from Eq. (2) and so on.