AI 中文总结
本文研究非相干测量下未知态纯度的乘法误差估计，提出非自适应与自适应算法，并证明其拷贝复杂度达到最优Θ(min{d/ε²+d^{3/2}/ε, d²/ε+√d/ε²})。
AI 中文摘要
在本工作中，我们考虑未知态纯度估计这一基本任务，要求达到乘法误差ε。当可以进行一般集体测量时，已知该问题需要Θ(√d/ε² + d/ε)份拷贝是必要且充分的[AISW20]。然而，在如此大量的拷贝上实现集体测量在实验上可能非常困难，因此我们旨在刻画在非相干测量下该问题的拷贝复杂度。在此设定下，唯一非平凡的结果是一个非自适应算法，使用O(d/ε² + d²/ε)份拷贝[PTTW26]，这比已知最佳下界多项式地更大。我们的第一个结果是一个新的非自适应非相干测量算法，使用O(d/ε² + d^{3/2}/ε)份拷贝即可成功，改进了拷贝复杂度中的后一项。此外，我们证明对于任何限制为非自适应测量的算法，上述拷贝复杂度是最优的。我们还开发了一种新的自适应纯度估计器，在高精度区域（即ε = o(1/d)）改进了上述复杂度。我们还表明，非自适应和自适应估计器一起产生了非相干纯度估计的最优复杂度；特别地，我们证明该问题的拷贝复杂度为Θ(min{d/ε² + d^{3/2}/ε, d²/ε + √d/ε²})。
英文摘要
In this work, we consider the fundamental task of estimating the purity of an unknown state to within $\textit{multiplicative}$ error $\varepsilon$. When one can perform general collective measurements, $Θ\left(\frac{\sqrt{d}}{\varepsilon^2} + \frac{d}{\varepsilon}\right)$ copies are known to be necessary and sufficient for this problem [AISW20]. However, implementing collective measurements on such a large number of copies can be experimentally demanding, and we thus aim to characterize the copy complexity of this problem with incoherent measurements. In this setting, the only non-trivial result is a non-adaptive algorithm that uses $O\left( \frac{d}{\varepsilon^2} + \frac{d^2}{\varepsilon} \right)$ copies [PTTW26], which is polynomially larger than the best-known lower bound. Our first result is a new algorithm performing non-adaptive incoherent measurements that succeeds using $O\left( \frac{d}{\varepsilon^2} + \frac{d^{3/2}}{\varepsilon} \right)$ copies, improving on the latter term in the copy complexity. Moreover, we show that for any algorithm restricted to non-adaptive measurements, the above copy complexity is optimal. We also develop a new adaptive estimator for the purity of a state that improves on the above complexity in the high-precision regime, i.e., for $\varepsilon = o(1/d)$. We also show that the non-adaptive and adaptive estimators together yield the optimal complexity for incoherent purity estimation; in particular, we show that the copy complexity of this problem is $$ Θ\left(\min\left\{ \frac{d}{\varepsilon^2} + \frac{d^{3/2}}{\varepsilon}, \frac{d^2}{\varepsilon} + \frac{\sqrt{d}}{\varepsilon^2}\right\} \right). $$
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