  How to generate genuine quantum randomness from untrusted devices is an
important problem in quantum information processing. Inspired by previous work
on a self-testing quantum random number generator [T. Lunghi et al., Phys. Rev.
Lett. 114, 150501 (2015)], we present a method to generate quantum randomness
from a prepare-and-measure scenario with independent devices. In existing
protocols, the quantum randomness depends only on a witness value (e.g.,
Clauser-Horne-Shimony-Holt value), which is calculated with the observed
probabilities. Differently, here all the observed probabilities are directly
used to calculate the min-entropy in our method. Through numerical simulation,
we find that the min-entropy of our proposed scheme is higher than that in the
previous work when a typical untrusted Bennett-Brassard 1984 (BB84) setup is
used. Consequently, thanks to the proposed method, more genuine quantum random
numbers may be obtained than before.
