Extraction of quantum randomness

Renato Renner

Yevgeniy Dodis, Renato Renner

Masahito Hayashi, Huangjun Zhu

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 R\'enyi 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.

Scott Aaronson

Kriss Gutierrez Anco, Tristan Nemoz, Peter J. Brown

How much cryptographically-secure randomness can be extracted from a quantum state? This fundamental question probes the absolute limits of quantum random number generation (QRNG) and yet, despite the technological maturity of QRNGs, it remains unsolved. In this work we consider a general adversarial model that allows for an adversary who has quantum side-information about both the source and the measurement device. Using links between randomness extraction rates and sandwiched Rényi entropies, we provide compact, easy to compute, achievable rates of secure randomness extraction from quantum states. In turn, this provides a simple to evaluate benchmarking tool for the randomness generation rates of QRNG protocols.