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Bri\"et et al. showed that an efficient communication protocol implies a reliable XOR game protocol. In this work, we improve this relationship, and obtain a nontrivial lower bound $2\log3\approx 3.1699$ of XOR-amortized communication complexity of the equality function. The proof uses an elegant idea of Paw\l owski et al. in a paper on information causality. Although the improvement of a lower bound of a communication complexity is at most a factor 2, all arguments and proofs in this work are quite simple and intuitive.

Better protocol for XOR game using communication protocol and nonlocal boxes