  Device-independent certification, also known as self-testing, aims at
guaranteeing the proper functioning of untrusted and uncharacterized devices.
For example, the quality of an unknown source expected to produce two-qubit
maximally entangled states can be evaluated in a bi-partite scenario, each
party using two binary measurements. The most robust approach consists in
deducing the fidelity of produced states with respect to a two-qubit maximally
entangled state from the violation of the CHSH inequality. In this paper, we
show how the self-testing of two-qubit maximally entangled states is improved
by a refined analysis of measurement statistics. The use of suitably chosen
Bell tests, depending on the observed correlations, allows one to conclude
higher fidelities than ones previously known. In particular, nontrivial
self-testing statements can be obtained from correlations that cannot be
exploited by a CHSH-based self-testing strategy. Our results not only provide
novel insight into the set of quantum correlations suited for self-testing, but
also facilitate the experimental implementations of device-independent
certifications.
