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 $\varepsilon, \zeta > 0$ and any $k\geq1$, we show that \[ \mathrm{Q}^1_{1-(1-\varepsilon)^{\Omega(\zeta^6k/\log|\mathcal{Z}|)}}(f^k) = \Omega\left(k\left(\zeta^5\cdot\mathrm{Q}^1_{\varepsilon + 12\zeta}(f) - \log\log(1/\zeta)\right)\right),\] 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, Pereszl\'{e}nyi 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 $\zeta$, then \[ \omega^*(G^k) = \left(1 - (1-\omega^*(G))^5\right)^{\Omega\left(\frac{\zeta^2 k}{\log(|\mathcal{A}|\cdot|\mathcal{B}|)}\right)}\] where $\omega^*(G)$ 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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单向量子通信的直接乘积定理

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 $\varepsilon, \zeta > 0$ 和任何 $k\geq1$，我们证明 \[ \mathrm{Q}^1_{1-(1-\varepsilon)^{\Omega(\zeta^6k/ \log|\mathcal{Z}|)}}(f^k) = \Omega\left(k\left(\zeta^5\cdot\mathrm{Q}^1_{\varepsilon + 12\zeta}(f ) - \log\log(1/\zeta)\right)\right),\] 其中 $\mathrm{Q}^1_{\varepsilon}(f)$ 表示单向纠缠辅助量子通信复杂度带有最坏情况错误 $\varepsilon$ 和 $f^k$ 的 $f$ 表示 $f$ 的 $k$ 个并行实例。据我们所知，这是量子通信的第一个直积定理。我们的技术受到两个玩家非本地博弈的纠缠值的平行重复定理的启发，在 Jain、Pereszl\'{e}nyi 和 Yao 的产品分布下，以及由于 Bavarian、Vidick 和 Yuen 的锚定分布，以及由于 Jain、Radhakrishnan 和 Sen 的量子协议的消息压缩。我们的技术也有效用于输入分布锚定在任何一侧的纠缠非本地游戏。特别是，我们证明对于任何游戏 $G = (q, \mathcal{X}\times\mathcal{Y}, \mathcal{A}\times\mathcal{B}, \mathsf{V})$ 其中 $ q$ 是 $\mathcal{X}\times\mathcal{Y}$ 上的分布，以锚定概率 $\zeta$ 锚定在任何一侧，然后 \[ \omega^*(G^k) = \left(1 - (1-\omega^*(G))^5\right)^{\Omega\left(\frac{\zeta^2 k}{\log(|\mathcal{A}|\cdot|\mathcal{ B}|)}\right)}\] 其中 $\omega^*(G)$ 表示博弈 $G$ 的纠缠值。这是 Bavarian、Vidick 和 Yuen 结果的概括，