A robust mathematical model for a loophole-free Clauser–Horne experiment

Abstract

Recent experiments (Giustina et al 2013 Nature 497 227–30; Christensen et al 2013 Phys. Rev. Lett. 111 130406) have reached detection efficiencies sufficient to close the detection loophole, testing the Clauser–Horne version of Bell's inequality. For a similar future experiment to be completely loophole-free, it will be important to have discrete experimental trials with randomized measurement settings for each trial, and the statistical analysis should not overlook the possibility of a local state varying over time with possible dependence on earlier trials (the ‘memory loophole’). In this paper, a mathematical model for such an experiment is presented, and a method for statistical analysis that is robust to memory effects is introduced. Additionally, a new method for calculating exact p-values for martingale-based statistics is described; previously, only non-sharp upper bounds derived from the Azuma–Hoeffding inequality have been available for such statistics. This improvement decreases the required number of experimental trials to demonstrate non-locality. The statistical techniques are applied to the data of Giustina et al (2013 Nature 497 227–30) and Christensen et al (2013 Phys. Rev. Lett. 111 130406) and found to perform well.