Title: Secure uniform random number extraction via incoherent strategies,
Abstract: To guarantee the security of uniform random numbers generated by a quantum
random number generator, we study secure extraction of uniform random numbers
when the environment of a given quantum state is controlled by the third party,
the eavesdropper. Here we restrict our operations to incoherent strategies that
are composed of the measurement on the computational basis and incoherent
operations (or incoherence-preserving operations). We show that the maximum
secure extraction rate is equal to the relative entropy of coherence. By
contrast, the coherence of formation gives the extraction rate when a certain
constraint is imposed on eavesdropper's operations. The condition under which
the two extraction rates coincide is then determined. Furthermore, we find that
the exponential decreasing rate of the leaked information is characterized by
Rényi relative entropies of coherence. These results clarify the power of
incoherent strategies in random number generation, and can be applied to
guarantee the quality of random numbers generated by a quantum random number
generator.