Pseudo-random number generators are widely used in many branches of science, mainly in applications related to Monte Carlo methods, although they are deterministic in design and, therefore, unsuitable for tackling fundamental problems in security and cryptography. The natural laws of the microscopic realm provide a fairly simple method to generate non-deterministic sequences of random numbers, based on measurements of quantum states. In practice, however, the experimental devices on which quantum random number generators are based are often unable to pass some tests of randomness. In this review, we briefly discuss two such tests, point out the challenges that we have encountered in experimental implementations and finally present a fairly simple method that successfully generates non-deterministic maximally random sequences.

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Random number generation plays an essential role in technology with important applications in areas ranging from cryptography to Monte Carlo methods, and other probabilistic algorithms. All such applications require high-quality sources of random numbers, yet effective methods for assessing whether a source produce truly random sequences are still missing. Current methods either do not rely on a formal description of randomness (NIST test suite) on the one hand, or are inapplicable in principle (the characterization derived from the Algorithmic Theory of Information), on the other, for they require testing all the possible computer programs that could produce the sequence to be analysed. Here we present a rigorous method that overcomes these problems based on Bayesian model selection. We derive analytic expressions for a model’s likelihood which is then used to compute its posterior distribution. Our method proves to be more rigorous than NIST’s suite and Borel-Normality criterion and its implementation is straightforward. We applied our method to an experimental device based on the process of spontaneous parametric downconversion to confirm it behaves as a genuine quantum random number generator. As our approach relies on Bayesian inference our scheme transcends individual sequence analysis, leading to a characterization of the source itself.

We present a complete theory for the full particle statistics of the positions of bulk and extremal particles in a one-dimensional Coulomb gas (CG) with an arbitrary potential, in the typical and large deviations regimes. Typical fluctuations are described by a universal function which depends solely on the general properties of the external potential. The rate function controlling large deviations is, rather unexpectedly, not strictly convex and has a discontinuous third derivative around its minimum for both extremal and bulk particles. This implies, in turn, that the rate function cannot predict the anomalous scaling of the typical fluctuations with the system size for bulk particles, and it may indicate the existence of an intermediate phase in this case. Moreover, its asymptotic behavior for extremal particles differs from the predictions of the Tracy-Widom distribution. Thus many of the paradigmatic properties of the full particle statistics of Dyson log gases do not carry over into their one-dimensional counterparts, hence proving that one-dimensional CG belongs to a different universality class. Our analytical expressions are thoroughly compared with Monte Carlo simulations, showing excellent agreement.

1. The authors define a function with value 1 for the positive examples and 0 for the negative ones. They fit a continuous function but do not deal at all with the error margin of the fit, which is almost as large as the function values they compute. 2. The term "quality" for the value of the fitted function gives the impression that some biological significance is associated with values of the fitted function strictly between 0 and 1, but there is no justification for this kind of interpretation and finding the point where the fit achieves its maximum does not make sense. 3. By neglecting the error margin the authors try to optimize the fitted function using differences in the second, third, fourth, and even fifth decimal place which have no statistical significance. 4. Even if such a fit could profit from more data points, the authors should first prove that the region of interest has some kind of smoothness, that is, that a continuous fit makes any sense at all. 5. "Simulated molecular evolution" is a misnomer. We are dealing here with random search. Since the margin of error is so large, the fitted function does not provide statistically significant information about the points in search space where strings with cleavage sites could be found. This implies that the method is a highly unreliable stochastic search in the space of strings, even if the neural network is capable of learning some simple correlations. 6. Classical statistical methods are for these kind of problems with so few data points clearly superior to the neural networks used as a "black box" by the authors, which in the way they are structured provide a model with an error margin as large as the numbers being computed.7. And finally, even if someone would provide us with a function which separates strings with cleavage sites from strings without them perfectly, so-called simulated molecular evolution would not be better than random selection.Since a perfect fit would only produce exactly ones or zeros,starting a search in a region of space where all strings in the neighborhood get the value zero would not provide any kind of directional information for new iterations. We would just skip from one point to the other in a typical random walk manner.