AI 中文总结
本文提出量子Rényi和Tsallis熵的近样本最优估计量，其样本复杂度优于现有估计量，匹配最新下界，推进了量子熵估计的样本复杂度研究。
AI 中文摘要
本文针对量子Rényi熵和Tsallis熵提供了具有近最优样本复杂度的估计量。具体而言，对于阶数α、维度d和加性误差ε：1. 当0<α<1时，Rényi熵的样本复杂度为O(d^(1+1/α)/ε^(1/α) + d^(1/α-1)/ε²)，Tsallis熵的样本复杂度为O(d^(1+1/α)/ε^(1/α) + d^(2-2α)/ε²)；特别地，当0<α≤1/2时，两种熵的样本复杂度均为O(d^(1+1/α)/ε^(1/α))。2. 当α>1且为非整数时，Rényi熵的样本复杂度为O(d²/ε^(1/α) + d^(1-1/α)/ε²)。本文的上界改进了Acharya等人（2017）提出的量子Rényi熵估计量，以及Chen、Liu和Wang（2026）提出的量子Tsallis熵估计量，且与Wang（2026）新近确立的下界匹配。
英文摘要
In this paper, we provide estimators for quantum Rényi and Tsallis entropies with nearly optimal sample complexity. Specifically, for order $α$, dimension $d$, and additive error $\varepsilon$, 1. For $0 < α< 1$, the sample complexity is $O(d^{1+1/α}/\varepsilon^{1/α} + d^{1/α-1}/\varepsilon^{2})$ for Rényi entropy and $O(d^{1+1/α}/\varepsilon^{1/α} + d^{2-2α}/\varepsilon^2)$ for Tsallis entropy. In particular, for $0 < α\leq 1/2$, the sample complexity for both entropies is $O(d^{1+1/α}/\varepsilon^{1/α})$. 2. For non-integer $α> 1$, the sample complexity is $O(d^2/\varepsilon^{1/α} + d^{1-1/α}/\varepsilon^2)$ for Rényi entropy. Our upper bounds improve the quantum Rényi entropy estimators due to Acharya, Issa, Shende, and Wagner (2017) and the quantum Tsallis entropy estimators due to Chen, Liu, and Wang (2026), and match the lower bounds recently established by Wang (2026).
Comments32 pages, 1 table, 4 algorithms
