Self-Testing of any Pure Entangled State with the Minimal Number of Measurements and Optimal Randomness Certification in a One-Sided Device-Independent Scenario

Certification of quantum systems and their properties has become a field of intensive study. Here, taking advantage of the one-sided device-independent (1SDI) scenario (also known as the quantum steering scenario), we propose a self-testing scheme for all bipartite entangled states using a single family of steering inequalities with the minimal number of two measurements per party. Building on this scheme we then show how to certify all rank-one extremal measurements, including nonprojective ${d}^{2}$-outcome measurements, which in turn can be used for certification of the maximal amount of randomness, that is, $2{\mathrm{log}}_{2}d$ bits. Finally, in the particular case of $d=3$, we propose an extended Bell scenario that transforms a 1SDI scheme to ``almost device independent.''