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arXiv · 2608.18070

Nearly Sample-Optimal Estimators for Quantum R\'enyi and Tsallis Entropies

Kean Chen·Qisheng Wang
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Abstract

In this paper, we provide estimators for quantum R\'enyi and Tsallis entropies with nearly optimal sample complexity. Specifically, for order $\alpha$, dimension $d$, and additive error $\varepsilon$, 1. For $0 < \alpha < 1$, the sample complexity is $O(d^{1+1/\alpha}/\varepsilon^{1/\alpha} + d^{1/\alpha-1}/\varepsilon^{2})$ for R\'enyi entropy and $O(d^{1+1/\alpha}/\varepsilon^{1/\alpha} + d^{2-2\alpha}/\varepsilon^2)$ for Tsallis entropy. In particular, for $0 < \alpha \leq 1/2$, the sample complexity for both entropies is $O(d^{1+1/\alpha}/\varepsilon^{1/\alpha})$. 2. For non-integer $\alpha > 1$, the sample complexity is $O(d^2/\varepsilon^{1/\alpha} + d^{1-1/\alpha}/\varepsilon^2)$ for R\'enyi entropy. Our upper bounds improve the quantum R\'enyi 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).

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Kean Chen, Qisheng Wang. 2026-08-18. Nearly Sample-Optimal Estimators for Quantum R\'enyi and Tsallis Entropies. https://arxiv.org/abs/2608.18070

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