Certifying the activation of Bell nonlocality with finite data

Abstract

The activation of Bell nonlocality is a protocol that enables the violation of a Bell inequality from a system that initially did not allow for any such violation because the state of the system was Bell-local. This activation of hidden Bell nonlocality has been demonstrated in experiments; however, while the certification of Bell nonlocality is conceptional straightforward, a statistically rigorous verification that the initial state is Bell-local has not yet been achieved. This is due to two key obstacles: The lack of a method to establish a suitable confidence region from the measured data and the need for an efficient technique to verify Bell locality of all states within the confidence region. In this work, we address both challenges. We introduce a confidence polytope in the form of a hyperoctahedron and provide a computationally efficient method to verify whether a quantum state admits a local hidden state model, thus being unsteerable and, consequently, Bell-local. Using these methods, we find that a statistically rigorous certification of hidden Bell nonlocality needs of the order of $10^9$ samples.