| Summary: | In order to reject the local hidden variables hypothesis, the usefulness of a Bell inequality can be quantified by how small a $p$-value it will give for a physical experiment. Here we show that to obtain a small expected $p$-value it is sufficient to have a large gap between the local and Tsirelson bounds of the Bell inequality, when it is formulated as a nonlocal game. We develop an algorithm for transforming an arbitrary Bell inequality into an equivalent nonlocal game with the largest possible gap, and show its results for the CGLMP and $I_{nn22}$ inequalities.
We present explicit examples of Bell inequalities with gap arbitrarily close to one, and show that this makes it possible to reject local hidden variables with arbitrarily small $p$-value in a single shot, without needing to collect statistics. We also develop an algorithm for calculating local bounds of general Bell inequalities which is significantly faster than the naïve approach, which may be of independent interest.
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