Optimal randomness certification in the quantum steering and prepare-and-measure scenarios

Quantum mechanics predicts the existence of intrinsically random processes. Contrary to classical randomness, this lack of predictability can not be attributed to ignorance or lack of control. Here we find the optimal method to quantify the amount of local or global randomness that can be extracted in two scenarios: (i) the quantum steering scenario, where two parties measure a bipartite system in an unknown state but one of them does not trust his measurement apparatus, and (ii) the prepare-and-measure scenario, where additionally the quantum state is known. We use our methods to compute the maximal amount of local and global randomness that can be certified by measuring systems subject to noise and losses and show that local randomness can be certified from a single measurement if and only if the detectors used in the test have detection efficiency higher than 50%.