The statistical strength of nonlocality proofs

Abstract

There exist numerous proofs of Bell's theorem, stating that quantum mechanics is incompatible with local realistic theories of nature. Here we define the strength of such nonlocality proofs in terms of the amount of evidence against local realism provided by the corresponding experiments. This measure tells us how many trials of the experiment we should perform in order to observe a substantial violation of local realism. Statistical considerations show that the amount of evidence should be measured by the Kullback-Leibler or relative entropy divergence between the probability distributions over the measurement outcomes that the respective theories predict. The statistical strength of a nonlocality proof is thus determined by the experimental implementation of it that maximizes the Kullback-Leibler divergence from experimental (quantum mechanical) truth to the set of all possible local theories. An implementation includes a specification with which probabilities the different measurement settings are sampled, and hence the maximization is done over all such setting distributions. We analyze two versions of Bell's nonlocality proof (his original proof and an optimized version by Peres), and proofs by Clauser-Horne-Shimony-Holt, Hardy, Mermin, and Greenberger-Horne-Zeilinger. We find that the GHZ proof is at least four and a half times stronger than all other proofs, while of the two-party proofs, the one of CHSH is the strongest.