Quantum Magic Rectangles: Characterization and Application to Certified Randomness Expansion

TL;DR

This paper generalizes the magic square game to rectangular dimensions, characterizes quantum strategies' success probabilities, and applies these results to quantum certified randomness expansion, analyzing noise tolerance and rates.

Contribution

It introduces a comprehensive analysis of rectangular magic games, characterizes their quantum winning probabilities, and applies these insights to improve quantum randomness expansion protocols.

Findings

Quantum strategies can win certain rectangular games with certainty.

For 1×n games, quantum strategies do not outperform classical strategies.

The paper provides bounds for 2×n games that outperform classical strategies.

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)].