Beyond the Magic Square Game: Widening the Gap for Two Bell States
Prior to this work, the largest known gap between the entangled value and the classical value for a one-round two-player nonlocal game with a perfect entangled strategy using two Bell states of entanglement was $\frac{1}{9}$, achieved by the Mermin-Peres magic square game. A larger gap of $\frac{2}{15}$ has been claimed for the related doily game of Kelleher, Roomy, and Holweck (Quant. Inf. Proc., 2024). However, this value assumes that the classical players use identical local strategies, and we show that dropping this assumption reduces the gap back to $\frac{1}{9}$. We then demonstrate that $\frac{1}{9}$ is not optimal by explicitly constructing a new nonlocal game with gap $\frac{4}{35}$.
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