# Secure uniform random number extraction via incoherent strategies

## TL;DR

This paper investigates secure extraction of uniform random numbers from quantum states using incoherent strategies, establishing the maximum extraction rate and analyzing the security under different constraints and measures.

## Contribution

It introduces the maximum secure extraction rate as the relative entropy of coherence and compares it with the coherence of formation under specific constraints.

## Key findings

- Maximum secure extraction rate equals the relative entropy of coherence.
- The coherence of formation determines the rate under certain eavesdropper constraints.
- The exponential decay of leaked information is characterized by Rényi relative entropies of coherence.

## 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 R\'{e}nyi 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.