On relating one-way classical and quantum communication complexities

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 x Y -> Z boolean OR {perpendicular to} be a partial function and mu be a distribution with support contained in f-1(Z). Denote d = |Z|. Let R-epsilon(1), (mu) ( f) be the classical one-way communication complexity of f; Q(epsilon)(1), (mu) (f) be the quantum one-way communication complexity of f and Q(epsilon)(1), mu, (f) be the entanglement-assisted quantum one-way communication complexity of f, each with distributional error (average error over mu) at most epsilon. We show: