On relating one-way classical and quantum communication complexities

Abstract

Communication complexity is the amount of communication needed to compute a function when the function inputs are distributed over multiple parties. In its simplest form, one-way communication complexity, Alice and Bob compute a function f(x,y), where x is given to Alice and y is given to Bob, and only one message from Alice to Bob is allowed. A fundamental question in quantum information is the relationship between one-way quantum and classical communication complexities, i.e., how much shorter the message can be if Alice is sending a quantum state instead of bit strings? We make some progress towards this question with the following results.Let f:X×Y→Z∪{⊥} be a partial function and μ be a distribution with support contained in f−1(Z). Denote d=|Z|. Let Rϵ1,μ(f) be the classical one-way communication complexity of f; Qϵ1,μ(f) be the quantum one-way communication complexity of f and Qϵ1,μ,∗(f) be the entanglement-assisted quantum one-way communication complexity of f, each with distributional error (average error over μ) at most ϵ. We show:1) If μ is a product distribution, η>0 and 0≤ϵ≤1−1/d, then,R2ϵ−dϵ2/(d−1)+η1,μ(f)≤2Qϵ1,μ,∗(f)+O(log⁡log⁡(1/η)).2)If μ is a non-product distribution and Z={0,1}, then ∀ϵ,η>0 such that ϵ/η+η<0.5,R3η1,μ(f)=O(Qϵ1,μ(f)⋅CS(f)/η3),whereCS(f)=maxyminz∈{0,1}|{x | f(x,y)=z}|.