# Tight Bounds for the Randomized and Quantum Communication Complexities of Equality with Small Error

We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability ϵ, 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/ϵ^2)+4)-cost private-coin communication protocol that computes the n-bit Equality function, to error ϵ. This improves upon the log(n/ϵ^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 loglog(1/ϵ)+O(1). In the quantum model, 1) we exhibit a one-way protocol of cost log(n/ϵ)+4, that uses only pure states and computes the n-bit Equality function to error ϵ. This bound was implicitly already shown by Nayak [PhD thesis'99]. 2) We show that any ϵ-error one-way protocol for n-bit Equality that uses only pure states communicates at least log(n/ϵ)-loglog(1/ϵ)-O(1) qubits. 3) We exhibit a one-way protocol of cost log(√(n)/ϵ)+3, that uses mixed states and computes the n-bit Equality function to error ϵ. This is also tight up to an additive loglog(1/ϵ)+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.

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