Generation of quantum randomness by probability estimation with classical side information

We develop the framework of probability estimation for certifying randomness with respect to classical side information from a sequence of Bell-test or other randomness-generating trials. The framework is based on directly estimating the probability of measurement outcomes conditional on settings choices and classical side information with adaptive test supermartingales. Accordingly, the number of trials needs not to be predetermined, and one can stop performing trials early, as soon as the desired amount of randomness is extractable. It can be used with arbitrary, partially known and time-varying probabilities for the random settings choices. It can also adapt to other time-varying experimental parameters. Furthermore, it is suitable for application to experiments with low Bell violation per trial, such as current optical loophole-free Bell tests. Compared with our previous work [Phys. Rev. A 98 040304(R) (2018)], here we formulate the framework for the general situation where the randomness can be extracted from a sequence of private data determined in an arbitrary way by the measurement outcomes of the trials. Trial-wise probability estimators can be adapted using all accessible, private information in addition to the results of previous trials. We prove that probability estimation achieves the asymptotically optimal rate for certified randomness generation and makes possible the exponential expansion of settings entropy. We implement probability estimation numerically and apply it to a representative settings-conditional distribution of the measurement outcomes from an atomic loophole-free Bell test [W. Rosenfeld et al., Phys. Rev. Lett. 119, 010402 (2017)] to illustrate trade-offs between the amount of randomness, error, settings entropy, adversarial settings bias, and number of trials. We then show that probability estimation yields more randomness from the optical loophole-free Bell-test data analyzed in [P. Bierhorst et al., arXiv:1702.05178v1] and tolerates adversarial settings biases.