Quantum Magic Rectangles: Characterisation and Application to Certified Randomness Expansion
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We study a generalisation of the Mermin-Peres magic square game to arbitrary rectangular dimensions. After exhibiting some general properties, these rectangular games are characterised in terms of their optimal win probabilities for quantum strategies. We find that for m × n rectangular games of dimensions m,n ≥ 3 there are quantum strategies that win with certainty, while for dimensions 1 × n quantum strategies do not outperform classical strategies. The final case of dimensions 2 × 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 by using our previous results and obtaining the winning probability of games with a distinguished input for which the devices give a deterministic outcome.
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