New bounds on classical and quantum one-way communication complexity

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 [KNR95] to include non-product distributions. Let ǫ ∈ (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 1,μ ǫ (f) = O((I(X : Y ) + 1) · VC(f)), where D ǫ (f) represents the one-way distributional communication complexity of f with error at most ǫ 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 ǫ (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 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 1,μ ǫ3/8 (f) = Ω(rec ǫ (f)), where Q ǫ3/8 (f) represents the quantum one-way distributional communication complexity of f with error at most ǫ/8 under μ and rec ǫ (f) represents the one-way rectangle bound of f with error at most ǫ under μ. Similarly for a non-boolean partial function f : X ×Y → Z∪{∗} and a product distribution μ on X × Y, we show, Q 1,μ ǫ6/(2·154) (f) = Ω(rec ǫ (f)). ⋆ School of Computer Science, and Institute for QuantumComputing, University of Waterloo, 200 University Ave. W., Waterloo, ON N2L 3G1, Canada. Research supported in part by ARO/NSA USA. ⋆⋆ Computer Science Department and Institute for Quantum Computing, California Institute of Technology, 1200 E California Bl, IQI, MC 107-81, Pasadena, CA 91125, USA. This work was supported by the National Science Foundation under grant PHY-0456720 and the Army Research Office under grant W911NF-05-10294 through Institute for Quantum Information at California Institute of Technology.