Three-input Majority Function is the Unique Optimal for the Bias Amplification using Nonlocal Boxes

Brassard et al. [Phys. Rev. Lett. 96, 250401 (2006)] showed that shared nonlocal boxes with the CHSH probability greater than 3+ √ 6 6 yields trivial communication complexity. There still exists the gap with the maximum CHSH probability 2+ √ 2 4 achievable by quantum mechanics. It is an interesting open question to determine the exact threshold for the trivial communication complexity. Brassard et al.’s idea is based on the recursive bias amplification by the 3-input majority function. It was not obvious if other choice of function exhibits stronger bias amplification. We show that the 3-input majority function is the unique optimal, so that one cannot improve the threshold 3+ √ 6 6 by Brassard et al.’s bias amplification.