Loophole-free Bell tests with randomly chosen subsets of measurement settings

There are bipartite quantum nonlocal correlations requiring very low detection efficiency to reach the loophole-free regime but that need too many measurement settings to be practical for actual experiments. This leads to the general problem of what can be concluded about loophole-free Bell nonlocality if only a random subset of these settings is tested. Here we develop a method to address this problem. We show that, in some cases, it is possible to detect loophole-free Bell nonlocality by testing only a small random fraction of the settings. The consequence is a higher detection efficiency. The method allows for the design of loophole-free Bell tests in which, given a quantum correlation that violates a Bell inequality, one can calculate the minimum fraction of contexts needed to reach the detection-loophole-free regime. The results also enforce a different way of thinking about how local realistic models or classical communication can be used to simulate quantum nonlocal correlations, as it shows that the amount of resources that are needed can be made arbitrarily large simply by considering more contexts.

Figure 1

Fraction of the settings required for a loophole-free test of the PNP CHSH Bell inequalities, as a function of $n$, the number of CHSH Bell inequalities conducted in parallel

Figure 2

Minimum detection efficiency ${\eta }_{\nu }$ needed to reach the loophole-free regime as a function of the fraction $\nu$ of the contexts for the PNP CHSH Bell inequality with different values of $n$. The threshold detection efficiencies for $n=10$, 11, 12, 13, and 14 are ${\eta }_{\mathrm{crit}}=0.43$, 0.38, 0.34, 0.31, and 0.28, respectively.