Testing randomness of series generated in Bell's experiment

The generation of series of random numbers is an important and difficult problem. Even the very definition of random is difficult. Appropriate measurements on entangled states have been proposed as the definitive solution to produce series of certified randomness. However, several reports indicate that quantum based devices show a disappointing rate of series rejected by standard tests of randomness. This problem is usually solved by using algorithms named extractors but, if the extractor were known by an eavesdropper (a situation that cannot be ruled out) the key security in QKD setups may be menaced. We use a toy fiber optic based setup, similar to a QKD one to be used in the field, to generate binary series, and evaluate their level of randomness according to Ville principle. Series are tested with a battery of standard statistical indicators, Hurst exponent, Kolmogorov complexity, minimum entropy, Takens dimension of embedding, and Augmented Dickey Fuller and Kwiatkowski Phillips Schmidt Shin to check stationarity. A theoretically predicted relationship between complexity and minimum entropy is observed. The good performance of a simple method to get useful series from rejected series, reported by Solis et al, is confirmed and supported with additional arguments. Regarding QKD, the level of randomness of series obtained by applying Toeplitz extractor to rejected series is found to be indistinguishable from the level of non-rejected raw ones.