发表机构
Center on Frontiers of Computing Studies, School of Computer Science, Peking University; School of Artificial Intelligence, Beijing Normal University; Mathematical Institute, University of Oxford（北京大学计算机科学学院前沿计算研究中心; 北京师范大学人工智能学院; 牛津大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出多级算法与变时估计，打破量子熵估计的乘法开销，将Tsallis、Rényi及冯·诺依曼熵的查询复杂度改进至近最优，并推广到更广泛的密度矩阵泛函。
AI 中文摘要
冯·诺依曼熵、Tsallis熵和Rényi熵是量子信息的基本度量。一种常见的估计策略将谱变换和统计读出的最坏情况成本相乘，导致与查询下界之间存在差距。我们通过使用局部归一化和精度分配的多级算法来减少这种开销，而变时估计则考虑了达到昂贵谱测试的概率。在受控纯化访问及其逆操作的条件下，我们获得了加性误差$\varepsilon$、秩上界$R$和固定阶$\alpha$的界限。当允许的维度和秩能容纳下界实例时，对于$1<\alpha<2$，Tsallis熵估计具有近最优的秩无关查询复杂度$\widetilde O(\varepsilon^{-1/(\alpha-1)})$；对于$\alpha\ge2$，复杂度为$\widetilde O(1/\varepsilon)$，其中$\widetilde O$抑制对数因子。前者节省了$1/\varepsilon$的因子；后者将已知的整数阶缩放扩展到非整数阶。我们还将冯·诺依曼熵的界限从$\widetilde O(R/\varepsilon^2)$改进到$\widetilde O(R/\varepsilon)$，并为非整数Rényi阶$1<\alpha<3$获得$\widetilde O(R/\varepsilon)$，在其他阶上还有进一步的界限。我们的泛函估计定理用函数幅度和谱质量加权的局部成本之和取代了全局乘积。我们进一步使用$\widetilde O(R/\varepsilon)$次查询估计固定的对数矩和熵方差，为有限块长量子压缩和纯态纠缠转换中的波动参数提供高效访问。更广泛地说，我们的框架超越了熵，扩展到一大类密度矩阵泛函，为量子谱估计中实现最优查询复杂度提供了一种系统方法。
英文摘要
The von Neumann, Tsallis, and Rényi entropies are fundamental measures of quantum information. A common estimation strategy multiplies the worst-case costs of spectral transformation and statistical readout, leaving gaps to query lower bounds. We reduce this overhead with multi-level algorithms that use local normalization and precision allocation, while variable-time estimation accounts for the probability of reaching expensive spectral tests. With controlled purified access and its inverse, we obtain bounds for additive error $\varepsilon$, rank upper bound $R$, and fixed order $α$. Tsallis entropy estimation has near-optimal rank-independent query complexity $\widetilde O(\varepsilon^{-1/(α-1)})$ for $1<α<2$, when the allowed dimension and rank accommodate the lower-bound instances, and $\widetilde O(1/\varepsilon)$ for $α\ge2$, where $\widetilde O$ suppresses logarithmic factors. The first saves a factor $1/\varepsilon$; the second extends known integer-order scaling to noninteger orders. We also improve the von Neumann entropy bound from $\widetilde O(R/\varepsilon^2)$ to $\widetilde O(R/\varepsilon)$ and obtain $\widetilde O(R/\varepsilon)$ for noninteger Rényi orders $1<α<3$, with further bounds at other orders. Our functional-estimation theorem replaces the global product by a sum of local costs weighted by function magnitudes and spectral masses. We further estimate fixed logarithmic moments and entropy variance using $\widetilde O(R/\varepsilon)$ queries, providing efficient access to fluctuation parameters that enter finite-blocklength quantum compression and pure-state entanglement conversion. More broadly, our framework extends beyond entropy to a broad class of density-matrix functionals, offering a systematic approach toward optimal query complexity in quantum spectral estimation.
CommentsMain text: 28 pages, 2 figures, 5 tables; appendices: 38 pages, 1 table. Total: 66 pages, 2 figures, 6 tables
