On relating one-way classical and quantum communication complexities
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Let f: X × Y →{0,1,} be a partial function and μ be a distribution with support contained in f^-1(0) ∪ f^-1(1). Let 𝖣^1,μ_ϵ(f) be the classical one-way communication complexity of f with average error under μ at most ϵ, 𝖰^1,μ_ϵ(f) be the quantum one-way communication complexity of f with average error under μ at most ϵ and 𝖰^1,μ, *_ϵ(f) be the entanglement assisted one-way communication complexity of f with average error under μ at most ϵ. We show: 1. If μ is a product distribution, then ∀ϵ, η > 0, 𝖣^1,μ_2ϵ + η(f) ≤𝖰^1,μ, *_ϵ(f) /η+O(log(𝖰^1,μ, *_ϵ(f))/η). 2. If μ is a non-product distribution, then ∀ϵ, η > 0 such that ϵ/η + η < 0.5, 𝖣^1,μ_3η(f) = O(𝖰^1,μ_ϵ(f) ·𝖢𝖲(f)/η^4), where 𝖢𝖲(f) = max_ymin_z∈{0,1}{|{x | f(x,y)=z}|}.
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