Secure uniform random-number extraction via incoherent strategies

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 the 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.

Figure 1

Extraction of secure uniform random numbers via incoherent strategy.

Figure 2

Extended strategy. Alice can apply a general incoherent (or incoherence-preserving) operation ${\mathrm{\Lambda }}_{\mathrm{i}}$. Both ${\mathcal{H}}_E$ and ${\mathcal{H}}_{E^{\prime }}$ are in Eve's hands.