量子SMP与单向通信的精确渐近速率和指数强逆
Exact Asymptotic Rates and an Exponential Strong Converse for quantum SMP and One-Way Communication
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中文总结 AI 辅助
本文证明量子SMP模型中联合计算多个实例的渐近最优通信速率等于行数和列数对数之和，并给出指数强逆，显示量子指纹优势在渐近下消失。
中文摘要 AI 辅助
联合计算多个实例可以减少每个实例的通信量。我们询问这种节省是否出现在同时消息传递（SMP）模型中，以及它们如何依赖于量子消息和共享资源。对于每个有限全函数 $f$ 和固定误差 $0\le\varepsilon<1$，我们证明 $f^n$ 的每实例最优最坏情况通信量收敛于 \\[ D^\parallel(f)=\log|\mathrm{Row}(f)|+\log|\mathrm{Col}(f)|, \\] 其中 $\mathrm{Row}(f)$ 和 $\mathrm{Col}(f)$ 是表 $(f(x,y))_{x,y}$ 的不同行和列的集合。这适用于没有共享纠缠的经典和量子SMP，无论有无共享随机性。因此，传输行和列索引是渐近最优的，即使有有界误差、联合计算和量子消息。特别是，在没有共享随机性的单个相等实例上，量子指纹的指数优势在渐近速率中消失。我们还证明了相同的阈值控制着一个指数强逆：每个低于它的固定最坏情况通信速率迫使正确计算所有 $n$ 个输出的概率以 $2^{-\Omega(n)}$ 衰减，即使在固定乘积输入分布下的平均情况下也是如此。使用共享随机性，最优期望通信速率为 $(1-\varepsilon)D^\parallel(f)$。每个发送方和裁判之间共享的纠缠将两个速率减半；任意的三方纠缠不会带来进一步的减少。结果扩展到一类全关系，并产生用 $\log|\mathrm{Row}(f)|$ 替换 $D^\parallel(f)$ 的单向特征。我们的证明将SMP简化为单向通信，并将成功计算与基于量子最小熵的采样联系起来；我们还使用pretty-good测量给出了关键引理的更简单证明。
英文摘要
Computing many instances jointly can reduce communication per instance. We ask whether such savings occur in the simultaneous-message-passing (SMP) model and how they depend on quantum messages and shared resources. For every finite total function $f$ and fixed error $0\le\varepsilon<1$, we prove that the optimal worst-case communication per instance for $f^n$ converges to \[ D^\parallel(f)=\log|\mathrm{Row}(f)|+\log|\mathrm{Col}(f)|, \] where $\mathrm{Row}(f)$ and $\mathrm{Col}(f)$ are the sets of distinct rows and columns of the table $(f(x,y))_{x,y}$. This holds for classical and quantum SMP without shared entanglement, with or without shared randomness. Thus, transmitting row and column indices is asymptotically optimal, even with bounded error, joint computation, and quantum messages. In particular, the exponential advantage of quantum fingerprinting for a single instance of equality without shared randomness disappears in the asymptotic rate. We also prove that the same threshold governs an exponential strong converse: every fixed worst-case communication rate below it forces the probability of computing all $n$ outputs correctly to decay as $2^{-Ω(n)}$, even on average under a fixed product input distribution. With shared randomness, the optimal expected communication rate is $(1-\varepsilon)D^\parallel(f)$. Entanglement shared between each sender and the referee halves both rates; arbitrary tripartite entanglement gives no further reduction. The results extend to a class of total relations and yield one-way characterizations with $D^\parallel(f)$ replaced by $\log|\mathrm{Row}(f)|$. Our proof reduces SMP to one-way communication and relates successful computation to sampling based on quantum min-entropy; we also give a simpler proof of the key lemma using the pretty-good measurement.
发表机构
- IQC, University of Waterloo（量子计算研究所，滑铁卢大学）
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