Regularising data for practical randomness generation

TL;DR

This paper introduces a method to optimize Bell inequality selection for certifying randomness in Bell experiments, improving min-entropy bounds by regularizing correlation estimates and tuning parameters based on numerical simulations.

Contribution

It proposes an efficient approach to choose Bell inequalities tailored to specific correlations, enhancing practical randomness certification.

Findings

Nearly always achieves higher min-entropy rates than standard CHSH inequality.

Regularising correlation estimates improves randomness certification.

Parameter tuning significantly affects min-entropy bounds.

Abstract

Non-local correlations that obey the no-signalling principle contain intrinsic randomness. In particular, for a specific Bell experiment, one can derive relations between the amount of randomness produced, as quantified by the min-entropy of the output data, and its associated violation of a Bell inequality. In practice, due to finite sampling, certifying randomness requires the development of statistical tools to lower-bound the min-entropy of the data as a function of the estimated Bell violation. The quality of such bounds relies on the choice of certificate, i.e., the Bell inequality whose violation is estimated. In this work, we propose a method for choosing efficiently such a certificate. It requires sacrificing a part of the output data in order to estimate the underlying correlations. Regularising this estimate then allows one to find a Bell inequality that is well suited for certifying practical randomness from these specific correlations. We then study the effects of various parameters on the obtained min-entropy bound and explain how to tune them in a favourable way. Lastly, we carry out several numerical simulations of a Bell experiment to show the efficiency of our method: we nearly always obtain higher min-entropy rates than when we use a pre-established Bell inequality, namely the Clauser-Horne-Shimony-Holt inequality.