Nearly Sample-Optimal Estimators for Quantum Rényi and Tsallis Entropies
2026-08-18 • Information Theory
Information Theory
AI summaryⓘ
The authors provide new methods to estimate quantum Rényi and Tsallis entropies efficiently, focusing on the number of samples needed to get accurate results. They give formulas for this sample complexity depending on parameters like order α, dimension d, and error tolerance ε. Their results improve on previous estimators and match known lower bounds, meaning their methods are close to the best possible. This helps in understanding quantum systems with fewer measurements.
Quantum entropyRényi entropyTsallis entropySample complexityQuantum measurementAdditive errorQuantum information theoryEstimatorDimensionOrder parameter
Authors
Kean Chen, Qisheng Wang
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).