Testing Quantumness via Photon Statistics for Time-Bin based Quantum Random Number Generators

Randomness is one of the essential components in many fields including cryptography and simulations. Several Quantum Random Number Generator (QRNG) models have been proposed to produce quantum random numbers, which, due to the quantum theory, are more secure than their classical counterparts. However, QRNGs can not produce true random numbers without deterministic classical post-processing. If the underlying distribution of the QRNG is close to a uniform distribution, a small amount of post-processing is sufficient to produce good random numbers retaining quantumness. In this work, we address the randomness and quantumness in the random numbers generated by the QRNGs. We consider two models of QRNGs, which ideally produce random numbers following different distributions (exponential and uniform), and show that, in practice, they are following similar distributions. These empirical photon distributions can be used to test the quantumness of a QRNG. In this letter, we suggest the χ² goodness-of-fit to test quantumness, as it is known to be an effective method to test if sample data follows a known distribution. We derive a relation when the underlying sampling distributions of the QRNGs will be ε-random. Depending on this relation, a suitable post-processing algorithm can be chosen.