积测试的最优分析
An Optimal Analysis of the Product Test
AI总结：
研究积测试的精确接受概率曲线，通过证明公式\(\mathrm{PT}_n(\omega)=\frac{1}{2}(1 + m\omega^2 + (1 - m\omega)^2)\)确定曲线，解决低重叠区域问题，改进哈罗 - 蒙塔纳罗约化的单次可靠性参数。
AI中文摘要：
积测试，即判定一个纯多体量子态在特定张量分解下是否完全无纠缠，是量子性质测试、无纠缠量子证明系统和张量优化之间的桥梁。尽管它是一个基本的性质测试任务且有诸多应用，但其精确（最坏情况）接受概率曲线尚未完全确定。本文精确确定了该曲线。设\(\omega\)为输入与积态的最大平方重叠，\(\mathrm{PT}_n(\omega)\)为积重叠为\(\omega\)的所有\(n\)体纯态上积测试的最大可能接受概率。证明了对于每个\(n\geq2\)和每个\(\omega\in(0,1]\)，\(\mathrm{PT}_n(\omega)=\frac{1}{2}(1 + m\omega^2 + (1 - m\omega)^2)\)，其中\(m=\lfloor1/\omega\rfloor\)。该公式恢复了\(\omega\geq1/2\)时曲线的已知紧区间，解决了所有低重叠区域\(\omega<1/2\)的情况，并表明\(\omega\to0\)时\(\mathrm{PT}_n(\omega)\to1/2\)。作为复杂度理论应用，结果改进了从\(\mathsf{QMA}(k)\)到\(\mathsf{QMA}(2)\)的哈罗 - 蒙塔纳罗约化中的单次可靠性参数。
英文摘要：
Product testing, i.e., deciding whether a pure multipartite quantum state is fully unentangled across a specified tensor decomposition, serves as a bridge between quantum property testing, unentangled quantum proof systems, and tensor optimization. Despite being a fundamental property testing task and having many applications, the product test's exact (worst-case) acceptance probability curve has yet to be fully determined. In this work, we determine this curve exactly. Let $ω$ be the maximum squared overlap of the input with a product state, and let $\mathrm{PT}_n(ω)$ be the largest possible acceptance probability of the product test over all $n$-partite pure states with product overlap $ω$, allowing arbitrary finite local dimensions. We prove that, for every $n\ge 2 $ and every $ω\in(0,1] $, $$ \mathrm{PT}_n(ω)=\frac12\left(1+mω^2+(1-mω)^2\right), $$ where $m=\lfloor1/ω\rfloor $. The formula recovers the previously known tight section of the curve for $ω\ge 1/2 $, resolves all low-overlap regimes $ω<1/2 $, and implies $\mathrm{PT}_n(ω)\to 1/2 $ as $ω\to 0$ answering an open problem in [Soleimanifar and Wright, SODA 2022]. As a complexity-theoretic application, our results improve the one-shot soundness parameter in the Harrow-Montanaro reduction from $\mathsf{QMA}(k)$ to $\mathsf{QMA}(2)$. Our techniques, built upon those of Soleimanifar and Wright, allow us to resolve these open questions while remaining surprisingly elementary.
