# Bell nonlocality with a single shot

Mateus Araújo1, Flavien Hirsch1, and Marco Túlio Quintino2,1,3

1Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
2Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
3Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan

### Abstract

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.

Nonlocal games are cooperative games between two parties, Alice and Bob, that are not allowed to communicate. The maximal probability with which Alice and Bob can win the game depends on how the world fundamentally works: if it respects classical ideas about locality and determinism, this maximal probability is given by the local bound. On the other hand, if the world works according to quantum mechanics, the maximal probability is given by the Tsirelson bound, which is larger than the local bound. This makes it possible to experimentally falsify the classical worldview: let Alice and Bob play a nonlocal game with quantum devices for many rounds, and if they win more often than the local bound predicts, that's it.

The number of rounds it takes for a decisive rejection of the classical worldview depends on the statistical power of the nonlocal game: a more powerful game requires fewer rounds to reach a conclusion with the same degree of confidence. We show that in order to get a large statistical power, it is enough to have a large gap between the local bound and the Tsirelson bound of the nonlocal game. Moreover, we show that this gap depends on how precisely a nonlocal game is formulated, so we develop an algorithm to maximise the gap over all possible formulations of a nonlocal game. With this, we derive the most powerful version of several well-known nonlocal games, such as the CHSH game, the CGLMP games, and the Inn22 games.

A natural question to ask is how high can the statistical power of a nonlocal game get. We show that it can get arbitrarily high, by constructing two nonlocal games with gap between their local and Tsirelson bounds arbitrarily close to one. This makes it possible to conclusively falsify the classical worldview with a single round of the nonlocal game, without needing to collect statistics. Unfortunately, neither of these games is experimentally feasible, so the question of whether a single-shot falsification is possible in practice is still open.

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### Cited by

[1] Mirjam Weilenmann, Edgar A. Aguilar, and Miguel Navascues, "Quantum Preparation Games", arXiv:2011.02216.

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