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Certification of quantum systems and their properties has become a field ofintensive studies. Here, taking advantage of the one-sided device-independentscenario (known also as quantum steering scenario), we propose a self-testingscheme for all bipartite entangled states using a single family of steeringinequalities with the minimal number of two measurements per party. Building onthis scheme we then show how to certify all rank-one extremal measurements,including non-projective $d^2$-outcome measurements, which in turn can be usedfor certification of the maximal amount of randomness from every entangledbipartite state of local dimension $d$, that is, $2\log_2d$ bits. Finally, in aparticular case of $d=3$, we extend our self-testing results to the fullydevice-independent setting.

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Self-testing of any pure entangled state with minimal number of  measurements and optimal randomness certification in one-sided  device-independent scenario