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).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).

quant-pharXiv:2608.18070Aug 19, 20260 citations

Generalized Schrödinger equation using Tsallis entropy

L. S. F. Olavo, Andris Figueiroa Bakuzis, R.Q. Amilcar

Physica A Statistical Mechanics and its ApplicationsSep 1, 199911 citations

Tsallis Entropy of Partitions in Quantum Logics

Abolfazl Ebrahimzadeh, Zahra Eslami Giski

International Journal of Theoretical PhysicsNov 26, 20182 citations

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).