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Winning a CHSH game without entangled particles in a finite number of biased rounds: how much luck is needed?

Gallus, C., Blasiak, P. & Pothos, E. M. ORCID: 0000-0003-1919-387X (2023). Winning a CHSH game without entangled particles in a finite number of biased rounds: how much luck is needed?. Entropy, 25(5), article number 824. doi: 10.3390/e25050824


Quantum games, like the CHSH game, are used to illustrate the puzzle and power of entanglement. These games are played over many rounds and in each round the participants, Alice and Bob, each receive a question bit to which they each have to give an answer bit, without being able to communicate during the game. When all possible classical answering strategies are analyzed, it is found that Alice and Bob cannot win more than 75% of the rounds. A higher percentage of wins arguably requiresCG an exploitable bias in the random generation of the questions bits or access to “non-local“ resources, like entangled pairs of particles. However, in an actual game the number of rounds has to be finite and question regimes may come up with unequal likelihood, so there always is a possibility for Alice and Bob to win by pure luck. This statistical possibility has to be transparently analyzed for practical applications such as the detection of eavesdropping in quantum communication. Similarly, when Bell tests are used in macroscopic situations to investigate the connection strength between system components and the validity of proposed causal models, the available data is limited and the possible combinations of question bits (measurement settings) may not be controlled to occur with equal likelihood. In the present work we give a fully self-contained proof for a bound on the probability to win a CHSH game by pure luck without making the usual assumption of only small biases in the random number generators. We also show bounds for the case of unequal probabilities based on results by McDiarmid and Combes and illustrate certain exploitable biases numerically.

Publication Type: Article
Additional Information: Copyright: c© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (
Publisher Keywords: CHSH Games; Bell Inequalities; Secure Quantum Communication; Quantum InformatioN
Subjects: Q Science > QA Mathematics
Departments: School of Health & Psychological Sciences > Psychology
SWORD Depositor:
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