单向量子通信的直积定理
A Direct Product Theorem for One-Way Quantum Communication
摘要：
我们证明了一般关系$f\subseteq\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}$的单向纠缠辅助量子通信复杂度的直积定理。对于任何$\varepsilon, ζ> 0$和任何$k\geq1$，我们证明了\[ \mathrm{Q}^1_{1-(1-\varepsilon)^{Ω(ζ^6k/\log|\mathcal{Z}|)}}(f^k) = Ω\left(k\left(ζ^5\cdot\mathrm{Q}^1_{\varepsilon + 12ζ}(f) - \log\log(1/ζ)\right)\right),\]，其中$\mathrm{Q}^1_{\varepsilon}(f)$表示$f$的单向纠缠辅助量子通信复杂度，具有最坏情况的错误$\varepsilon$和$f^k$表示$f$的$k$并行实例。据我们所知，这是量子通信的第一直积定理。我们的技术受到两个玩家非局部游戏纠缠值的并行重复定理的启发，根据Jain，Pereszlényi和Yao的产品分布，以及Bavarian，Vidick和Yuen的锚定分布，以及Jain的量子协议的消息压缩，Radhakrishnan和Sen.我们的技术也适用于纠缠非本地游戏，这些游戏的输入分布锚定在任何一边。特别是，我们表明，对于任何游戏$G = (q, \mathcal{X}\times\mathcal{Y}, \mathcal{A}\times\mathcal{B}, \mathsf{V})$，$q$是一个分布在$\mathcal{X}\times\mathcal{Y}$锚定在任何一侧的锚定概率$ζ$，然后\[ ω^*(G^k) = \left(1 - (1-ω^*(G))^5\right)^{Ω\left(\frac{ζ^2 k}{\log(|\mathcal{A}|\cdot|\mathcal{B}|)}\right)}\]，其中$ω^*(G)$表示纠缠值的游戏$G$。这是Bavarian、Vidick和Yuen的结果的推广，他们证明了锚定在双方的博弈的平行重复定理，并且可能是他们证明的简化。
英文摘要：
We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation $f\subseteq\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}$. For any $\varepsilon, ζ> 0$ and any $k\geq1$, we show that \[ \mathrm{Q}^1_{1-(1-\varepsilon)^{Ω(ζ^6k/\log|\mathcal{Z}|)}}(f^k) = Ω\left(k\left(ζ^5\cdot\mathrm{Q}^1_{\varepsilon + 12ζ}(f) - \log\log(1/ζ)\right)\right),\] where $\mathrm{Q}^1_{\varepsilon}(f)$ represents the one-way entanglement-assisted quantum communication complexity of $f$ with worst-case error $\varepsilon$ and $f^k$ denotes $k$ parallel instances of $f$. As far as we are aware, this is the first direct product theorem for quantum communication. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao, and under anchored distributions due to Bavarian, Vidick and Yuen, as well as message-compression for quantum protocols due to Jain, Radhakrishnan and Sen. Our techniques also work for entangled non-local games which have input distributions anchored on any one side. In particular, we show that for any game $G = (q, \mathcal{X}\times\mathcal{Y}, \mathcal{A}\times\mathcal{B}, \mathsf{V})$ where $q$ is a distribution on $\mathcal{X}\times\mathcal{Y}$ anchored on any one side with anchoring probability $ζ$, then \[ ω^*(G^k) = \left(1 - (1-ω^*(G))^5\right)^{Ω\left(\frac{ζ^2 k}{\log(|\mathcal{A}|\cdot|\mathcal{B}|)}\right)}\] where $ω^*(G)$ represents the entangled value of the game $G$. This is a generalization of the result of Bavarian, Vidick and Yuen, who proved a parallel repetition theorem for games anchored on both sides, and potentially a simplification of their proof.
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