On relating one-way classical and quantum communication complexities

Communication complexity is the amount of communication needed to compute a function when inputs are distributed over multiple parties. In its simplest form, one-way complexity, Alice and Bob f(x,y), where xmlns:mml="http://www.w3.org/1998/Math/MathML">x given xmlns:mml="http://www.w3.org/1998/Math/MathML">y Bob, only one message from allowed. A fundamental question in quantum information relationship between classical complexities, i.e., how much shorter can be if sending state instead bit strings? We make some progress towards this with following results.Let xmlns:mml="http://www.w3.org/1998/Math/MathML">f:X×Y→Z∪{⊥} partial xmlns:mml="http://www.w3.org/1998/Math/MathML">μ distribution support contained xmlns:mml="http://www.w3.org/1998/Math/MathML">f−1(Z). Denote xmlns:mml="http://www.w3.org/1998/Math/MathML">d=|Z|. Let xmlns:mml="http://www.w3.org/1998/Math/MathML">Rϵ1,μ(f) xmlns:mml="http://www.w3.org/1998/Math/MathML">f; mathvariant="sans-serif">Qϵf class="MJX-TeXAtom-ORD">1,μ,∗f, each distributional error (average xmlns:mml="http://www.w3.org/1998/Math/MathML">μ) at most xmlns:mml="http://www.w3.org/1998/Math/MathML">ϵ. show:1) If product distribution, xmlns:mml="http://www.w3.org/1998/Math/MathML">η>0 xmlns:mml="http://www.w3.org/1998/Math/MathML">0≤ϵ≤1−1/d, then,R2ϵ−dϵ2/(d−1)+η)≤2Qϵ)+O(log⁡log⁡(1/η)).2)If non-product xmlns:mml="http://www.w3.org/1998/Math/MathML">Z={0,1}, then xmlns:mml="http://www.w3.org/1998/Math/MathML">∀ϵ,η>0 such that xmlns:mml="http://www.w3.org/1998/Math/MathML">ϵ/η+η<0.5,3η)=O()⋅CS)/η3,where)=maxyminz∈}|{x | f)=z}|.