Nearly Sample-Optimal Estimators for Quantum Rényi and Tsallis Entropies

2026-08-18

AI Summary

本文提出了用于估计量子Rényi和Tsallis熵的几乎最优样本复杂度的估计器，改进了现有方法，并与最新下界相匹配。

Abstract

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).

Problem

• quantum Rén yi entropy
• Tsallis entropy
• sample complexity

Innovation

• Quantum Rényi Entropy
• Quantum Tsallis Entropy
• Sample Complexity
• Optimal Estimators