Abstract
In order to reject the local hidden variables hypothesis, the usefulness of a Bell inequality can be quantified by how small ap-value it will give for a physical experiment. Here we show that to obtain a small expectedp-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 andInn22inequalities.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 smallp-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.
Highlights
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
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
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 Inn22 inequalities
Summary
There are two main approaches to the study of Bell nonlocality. In the physics literature it is common to use Bell inequalities. As we show in Appendix B, if we allow lifting, i.e., embedding the Bell inequality in a scenario with more inputs, it is possible to transform all two-outcome Bell inequalities into nonlocal games with deterministic predicate. The local and Tsirelson bounds of a nonlocal game are physically meaningful, and more generally we refer always to the probability of winning the game, as opposed to obtaining some value in the left hand side of the Bell inequality. This is very convenient for statistical analysis, and makes comparison between different nonlocal games meaningful. It is easy to see that the ratio ωq(GKVd )/ω (GKVd ) grows without bound with d, but the upper bound we present goes to one
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