  Verifying entanglement between parties is essential for creating a secure
quantum network, and Bell tests are the most rigorous method for doing so.
However, if there is any signaling between the parties, then the violation of
these inequalities can no longer be used to draw conclusions about the presence
of entanglement. This is because signaling between the parties allows them to
coordinate their measurement settings and outcomes, which can give rise to a
violation of Bell inequalities even if the parties are not genuinely entangled.
There is a pressing need to examine the role of signaling in quantum
communication protocols from multiple perspectives, including communication
security, physics foundations, and resource utilization while also promoting
innovative technological applications. Here, we propose a semi-device
independent protocol that allows us to numerically correct for effects of
correlations in experimental probability distributions, caused by statistical
fluctuations and experimental imperfections. Our noise robust protocol presents
a relaxation of a tomography-based optimisation method called the steering
robustness, that uses semidefinite programming to numerically identify the
optimal quantum steering inequality without the need for resource-intensive
tomography. The proposed protocol is numerically and experimentally analyzed in
the context of random, misaligned measurements, correcting for signalling where
necessary, resulting in a higher probability of violation compared to existing
state-of-the-art inequalities. Our work demonstrates the power of semidefinite
programming for entanglement verification and brings quantum networks closer to
practical applications.
