Correcting for finite statistics effects in a quantum steering experiment

Verifying entanglement between parties is essential for creating secure quantum communication. However, finite statistics can lead to false positive outcomes in any tests for entanglement. Here, we introduce a one-sided device-independent protocol that corrects for apparent signaling effects in experimental probability distributions, caused by statistical fluctuations and experimental imperfections. We use semidefinite programming to identify the optimal inequality, for our experimental probability distribution, without resource-intensive tomography. Our protocol is numerically and experimentally analysed in the context of random, misaligned measurements, correcting apparent signaling where necessary. Our results show a significantly higher probability of violation than existing state-of-the-art inequalities. This work demonstrates the power of semidefinite programming for entanglement verification and brings quantum networks closer to practical applications.