We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $\epsilon$, getting optimal constant factors in the leading terms in a number of different models. In the randomized model, 1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newman's theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. 2) Using this we obtain a $(\log(n/\epsilon^2)+4)$-cost private-coin communication protocol that computes the $n$-bit Equality function, to error $\epsilon$. This improves upon the $\log(n/\epsilon^3)+O(1)$ upper bound implied by Newman's theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive $\log\log(1/\epsilon)+O(1)$. In the quantum model, 1) we exhibit a one-way protocol of cost $\log(n/\epsilon)+4$, that uses only pure states and computes the $n$-bit Equality function to error $\epsilon$. This bound was implicitly already shown by Nayak [PhD thesis'99]. 2) We show that any $\epsilon$-error one-way protocol for $n$-bit Equality that uses only pure states communicates at least $\log(n/\epsilon)-\log\log(1/\epsilon)-O(1)$ qubits. 3) We exhibit a one-way protocol of cost $\log(\sqrt{n}/\epsilon)+3$, that uses mixed states and computes the $n$-bit Equality function to error $\epsilon$. This is also tight up to an additive $\log\log(1/\epsilon)+O(1)$, which follows from Alon's result. 4) We study the number of EPR pairs required to be shared in an entanglement-assisted one-way protocol. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix.

翻译：我们研究了经典等式函数在小误差概率ε下的随机与量子通信复杂度，在若干不同模型中取得了领先项的最优常数因子。在随机模型方面：1）我们提出了一种通用技术，通过引入小乘法误差和极小加法开销，将公共硬币协议转换为私有硬币协议。与Newman定理[Inf. Proc. Let.'91]相比，该方法在误差参数依赖性上有所改进。2）利用该技术，我们获得了一个成本为$(\log(n/\epsilon^2)+4)$的私有硬币通信协议，能计算n比特等式函数并达到误差ε。这一结果改进了由Newman定理导出的$\log(n/\epsilon^3)+O(1)$上界，并与当前已知最佳下界（源自Alon [Comb. Prob. Comput.'09]）仅相差加法项$\log\log(1/\epsilon)+O(1)$。在量子模型方面：1）我们展示了一个成本为$\log(n/\epsilon)+4$、仅使用纯态的单向协议，能计算n比特等式函数并达到误差ε。该界已隐含在Nayak的博士论文[PhD thesis'99]中。2）我们证明任何仅使用纯态且误差为ε的n比特等式函数单向协议，至少需要通信$\log(n/\epsilon)-\log\log(1/\epsilon)-O(1)$量子比特。3）我们展示了一个成本为$\log(\sqrt{n}/\epsilon)+3$、使用混合态的单向协议，能计算n比特等式函数并达到误差ε。该结果同样紧于Alon结果导出的加法项$\log\log(1/\epsilon)+O(1)$。4）我们研究了纠缠辅助单向协议中所需共享的EPR对数量。我们的上界也给出了单位矩阵的近似秩及相关测度的上界。