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 sce-nario), 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 d2-outcome measure-ments, which in turn can be used for certification of the maximal amount of randomness, that is, 2 log2 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."