AI 中文总结
该研究针对d维量子态的频谱估计等任务，证明其样本复杂度下界为Ω(d²⁻γ)，通过构造Haar随机投影器夹积的困难实例，结合Jucys-Murphy元素等分析完成下界推导。
AI 中文摘要
我们研究未知量子态的基本酉不变性质估计与测试的样本复杂度，具体包括频谱估计、冯·诺依曼熵估计和秩检验任务。对于d维量子态，以及任意γ>0，我们证明了：频谱估计达到恒定排序总变差误差、熵估计达到恒定加性误差、秩检验达到恒定迹距离时，样本复杂度的下界为Ω(d²⁻γ)。我们的困难实例由Haar随机投影器的夹积构造，通过一种新技巧对其进行适当归一化，该技巧可推导所得态的高阶张量矩，这些矩可表示为对称群代数中Jucys-Murphy元素的对称函数。为证明两类混合态不可区分，我们分析对数似然比并执行矩匹配，即将其低阶Jucys-Murphy分量置零；随后通过高阶分量界定f散度以得到不可区分性，非零高阶项及Haar随机幺正函数的集中性也暗示了典型频谱、熵和秩的分离，从而证明了所有下界。
英文摘要
We study the sample complexity of estimating and testing fundamental unitarily invariant properties of unknown quantum states; namely, the tasks of spectrum estimation, von Neumann entropy estimation, and rank-testing. For $d$-dimensional states, and for every $γ>0$, we prove a sample complexity lower bound of $Ω(d^{2-γ})$ for spectrum estimation to constant sorted total-variation error, entropy estimation to constant additive error, and rank-testing to constant trace distance. Our hard instances are constructed from sandwiched products of Haar-random projectors, suitably normalized using a novel technique that lets us derive explicit expressions for high-order tensor moments of the resultant states. These moments can be expressed as symmetric functions of Jucys--Murphy elements of the symmetric group algebra. To show that two such mixtures are indistinguishable, we analyze the log-likelihood ratio and perform moment-matching, i.e., we set its low-order Jucys--Murphy components to zero. Indistinguishability is then obtained by bounding an $f$-divergence through the high-order components; the non-zero high-order terms and concentration of functions of Haar-random unitaries also imply separations in typical spectra, entropies, and ranks, proving all our lower bounds.
Comments48 pages
