We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $\epsilon$, getting the 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. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix. This also implies improved upper bounds on these measures for the distributed SINK function, which was recently used to refute the randomized and quantum versions of the log-rank conjecture.

翻译：我们用小差错概率 $\ epsilon 来调查受广泛研究的平等功能的随机和量子通信复杂性。 在随机模型中, 1) 我们给普通技术, 将公共coin协议转换为私人coin协议, 产生一个小的重复性错误, 成本很小。 这是对错误参数依赖的Newman 理论[Inf. Proc. Let.'91] 的改进。 2 利用这个方法, 我们获得了美元(n) (n/ epsilon$ ) 的优化常数。 美元($) 美元(lision) + 美元(lus- 美元) 。 美元(l) 美元(lision_ 美元) 的内存协议使用美元( 美元) 美元(lision_ ) 的上限值。 由 Newman 的元(ncorisal_ commission) 的内存( web. comb.