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

Renyi Entropy and the Uncertainty Relations

Iwo Białynicki‐Birula

AIP conference proceedingsJan 1, 200743 citations

Conditional Rényi Entropy

Marco Tomamichel

SpringerBriefs in mathematical physicsOct 8, 20154 citations

Generalized statistical complexity and Fisher-Renyi entropy product in the $H$ -atomGeneralized statistical complexity and Fisher-Renyi entropy product in the $H$-atom

E. Romera, Ricardo López‐Ruiz, Jaime Sañudo A +1 more

By using the Renyi entropy, and following the same scheme that in the Fisher-Renyi entropy product case, a generalized statistical complexity is defined. Several properties of it, including inequalities and lower and upper bounds are derived. The hydrogen atom is used as a test system where to quantify these two different statistical magnitudes, the Fisher-Renyi entropy product and the generalized statistical complexity. For each level of energy, both indicators take their minimum values on the orbitals that correspond to the highest orbital angular momentum. Hence, in the same way as happens with the Fisher-Shannon and the statistical complexity, these generalized Renyi-like statistical magnitudes break the energy degeneration in the H-atom.By using the Renyi entropy, and following the same scheme that in the Fisher-Renyi entropy product case, a generalized statistical complexity is defined. Several properties of it, including inequalities and lower and upper bounds are derived. The hydrogen atom is used as a test system where to quantify these two different statistical magnitudes, the Fisher-Renyi entropy product and the generalized statistical complexity. For each level of energy, both indicators take their minimum values on the orbitals that correspond to the highest orbital angular momentum. Hence, in the same way as happens with the Fisher-Shannon and the statistical complexity, these generalized Renyi-like statistical magnitudes break the energy degeneration in the H-atom.

arXiv:0901.1752Jan 13, 20098 citations

Smooth Entropy and Rényi Entropy

Christian Cachin

Lecture notes in computer scienceJan 1, 199739 citations

Properties of Rényi complexity ratio of quantum states for central potential

Debraj Nath

Rényi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of Rényi complexity ratio are proved by Lebesgue measure. Some properties of Rényi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of Rényi entropy, Rényi complexity ratio, statistical complexities based on Rényi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of Rényi complexity ratio are verified for some diatomic molecules (CO, NO, N $_2$ , CH, H $_2$ , and ScH) and for some other quantum systems.Rényi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of Rényi complexity ratio are proved by Lebesgue measure. Some properties of Rényi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of Rényi entropy, Rényi complexity ratio, statistical complexities based on Rényi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of Rényi complexity ratio are verified for some diatomic molecules (CO, NO, N$_2$, CH, H$_2$, and ScH) and for some other quantum systems.

arXiv:2008.05418Aug 12, 20201 citation

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).

Generalized statistical complexity and Fisher-Renyi entropy product in the

H

-atom

_2

, CH, H

_2

, and ScH) and for some other quantum systems.