Abstract
We study a generalization of the Mermin-Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these rectangular games are fully characterized in terms of their optimal win probabilities for quantum strategies. We find that for $m \times n$ rectangular games of dimensions $m,n \geq 3$ there are quantum strategies that win with certainty, while for dimensions $1 \times n$ quantum strategies do not outperform classical strategies. The final case of dimensions $2 \times n$ is richer, and we give upper and lower bounds that both outperform the classical strategies. Finally, we apply our findings to quantum certified randomness expansion to find the noise tolerance and rates for all magic rectangle games. To do this, we use our previous results to obtain the winning probability of games with a distinguished input for which the devices give a deterministic outcome, and follow the analysis of C. A. Miller and Y. Shi [SIAM J. Comput. 46, 1304 (2017)].
Highlights
Quantum theory has been arguably one of the most successful scientific theories, especially in terms of accuracy of predictions and applications
We introduce a new class of nonlocal games which we call magic rectangle games, characterize their winning probabilities, and apply the results to certified randomness expansion
Using the known fact that the regular magic square game can be won for quantum strategies with certainty, we reduce the full characterization of magic rectangles to that of 1 × n and 2 × n games (Theorem 2)
Summary
Quantum theory has been arguably one of the most successful scientific theories, especially in terms of accuracy of predictions and applications. We get that 2 × n games with n 3 can be won with certainty using behaviors at level 1 of the NPA hierarchy (and so exhibit a version of “pseudotelepathy”), while the quantum and almost quantum sets both give winning probabilities strictly smaller than unity ( not exhibiting pseudotelepathy) We use this characterization to analyze certified randomness expansion from magic rectangle games. Arnon-Friedman et al [29], Brown et al [30] detail alternative techniques, which give better rates for the spot-checking protocol by using the entropy accumulation theorem [31,32] These are more involved and case-specific than [14] and, to give a general analysis of certified randomness for all magic rectangle games, we use [14] in our work. VII where we discuss our results and give future directions
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