New bounds on classical and quantum one-way communication complexity

Abstract

In this paper we provide new bounds on classical and quantum distributional communication complexity in the two-party, one-way model of communication. In the classical one-way model, our bound extends the well known upper bound of Kremer, Nisan and Ron [I. Kremer, N. Nisan, D. Ron, On randomized one-round communication complexity, in: Proceedings of The 27th ACM Symposium on Theory of Computing, STOC, 1995, pp. 596–605] to include non-product distributions. Let ϵ ∈ ( 0 , 1 / 2 ) be a constant. We show that for a boolean function f : X × Y → { 0 , 1 } and a non-product distribution μ on X × Y , D ϵ 1 , μ ( f ) = O ( ( I ( X : Y ) + 1 ) ⋅ VC ( f ) ) , where D ϵ 1 , μ ( f ) represents the one-way distributional communication complexity of f with error at most ϵ under μ ; VC ( f ) represents the Vapnik–Chervonenkis dimension of f and I ( X : Y ) represents the mutual information, under μ , between the random inputs of the two parties. For a non-boolean function f : X × Y → { 1 , … , k } ( k ≥ 2 an integer), we show a similar upper bound on D ϵ 1 , μ ( f ) in terms of k , I ( X : Y ) and the pseudo-dimension of f ′ = def f k , a generalization of the VC -dimension for non-boolean functions. In the quantum one-way model we provide a lower bound on the distributional communication complexity, under product distributions, of a function f , in terms of the well studied complexity measure of f referred to as the rectangle bound or the corruption bound of f . We show for a non-boolean total function f : X × Y → Z and a product distribution μ on X × Y , Q ϵ 3 / 8 1 , μ ( f ) = Ω ( rec ϵ 1 , μ ( f ) ) , where Q ϵ 3 / 8 1 , μ ( f ) represents the quantum one-way distributional communication complexity of f with error at most ϵ 3 / 8 under μ and rec ϵ 1 , μ ( f ) represents the one-way rectangle bound of f with error at most ϵ under μ . Similarly for a non-boolean partial function f : X × Y → Z ∪ { ∗ } and a product distribution μ on X × Y , we show, Q ϵ 6 / ( 2 ⋅ 1 5 4 ) 1 , μ ( f ) = Ω ( rec ϵ 1 , μ ( f ) ) .