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 $f: \mathcal{X} \times \mathcal{Y} \rightarrow \mathcal{Z} \cup \{\bot\}$ be a partial function and $\mu$ be a distribution with support contained in $f^{-1}(\mathcal{Z})$. Denote $d=|\mathcal{Z}|$. Let $\mathsf{R}^{1,\mu}_\epsilon(f)$ be the classical one-way communication complexity of $f$; $\mathsf{Q}^{1,\mu}_\epsilon(f)$ be the quantum one-way communication complexity of $f$ and $\mathsf{Q}^{1,\mu, *}_\epsilon(f)$ be the entanglement-assisted quantum one-way communication complexity of $f$, each with distributional error (average error over $\mu$) at most $\epsilon$. We show: 1) If $\mu$ is a product distribution, $\eta > 0$ and $0 \leq \epsilon \leq 1-1/d$, then, $$\mathsf{R}^{1,\mu}_{2\epsilon -d\epsilon^2/(d-1)+ \eta}(f) \leq 2\mathsf{Q}^{1,\mu, *}_{\epsilon}(f) + O(\log\log (1/\eta))\enspace.$$ 2)If $\mu$ is a non-product distribution and $\mathcal{Z}=\{ 0,1\}$, then $\forall \epsilon, \eta > 0$ such that $\epsilon/\eta + \eta < 0.5$, $$\mathsf{R}^{1,\mu}_{3\eta}(f) = O(\mathsf{Q}^{1,\mu}_{{\epsilon}}(f) \cdot \mathsf{CS}(f)/\eta^3)\enspace,$$ where \[\mathsf{CS}(f) = \max_{y} \min_{z\in\{0,1\}} \vert \{x~|~f(x,y)=z\} \vert \enspace.\]

翻译：通信复杂度是指当函数输入分布于多个参与方时，计算该函数所需的通信量。其最简单形式为单向通信复杂度：Alice和Bob计算函数$f(x,y)$，其中$x$赋予Alice、$y$赋予Bob，且仅允许从Alice向Bob发送一条消息。量子信息中的一个基本问题是单向量子与经典通信复杂度之间的关系，即当Alice发送量子态而非比特串时，消息可以缩短多少？我们通过以下结果在此问题上取得进展。设$f: \mathcal{X} \times \mathcal{Y} \rightarrow \mathcal{Z} \cup \{\bot\}$为部分函数，$\mu$为支撑集包含于$f^{-1}(\mathcal{Z})$的分布，记$d=|\mathcal{Z}|$。令$\mathsf{R}^{1,\mu}_\epsilon(f)$为$f$的经典单向通信复杂度；$\mathsf{Q}^{1,\mu}_\epsilon(f)$为$f$的量子单向通信复杂度；$\mathsf{Q}^{1,\mu, *}_\epsilon(f)$为$f$的纠缠辅助量子单向通信复杂度，各量在分布误差（关于$\mu$的平均误差）不超过$\epsilon$的条件下定义。我们证明：1) 若$\mu$为乘积分布，$\eta > 0$且$0 \leq \epsilon \leq 1-1/d$，则$$\mathsf{R}^{1,\mu}_{2\epsilon -d\epsilon^2/(d-1)+ \eta}(f) \leq 2\mathsf{Q}^{1,\mu, *}_{\epsilon}(f) + O(\log\log (1/\eta))\enspace.$$ 2) 若$\mu$为非乘积分布且$\mathcal{Z}=\{ 0,1\}$，则对任意满足$\epsilon/\eta + \eta < 0.5$的$\epsilon, \eta > 0$有$$\mathsf{R}^{1,\mu}_{3\eta}(f) = O(\mathsf{Q}^{1,\mu}_{{\epsilon}}(f) \cdot \mathsf{CS}(f)/\eta^3)\enspace,$$其中\[\mathsf{CS}(f) = \max_{y} \min_{z\in\{0,1\}} \vert \{x~|~f(x,y)=z\} \vert \enspace.\]