Relative entropy of coherence can be written as an entropy difference of the original state and the incoherent state closest to it when measured by relative entropy. The natural question is, if we generalize this situation to Tsallis or Rényi entropies, would it define good coherence measures? In other words, we define a difference between Tsallis entropies of the original state and the incoherent state closest to it when measured by Tsallis relative entropy. Taking Rényi entropy instead of the Tsallis entropy, leads to the well-known distance-based Rényi coherence, which means this expression defined a good coherence measure. Interestingly, we show that Tsallis entropy does not generate even a genuine coherence monotone, unless it is under a very restrictive class of operations. Additionally, we provide continuity estimates for both Tsallis and Rényi coherence expressions. Furthermore, we present two coherence measures based on the closest incoherent state when measures by Tsallis or Rényi relative entropy.Relative entropy of coherence can be written as an entropy difference of the original state and the incoherent state closest to it when measured by relative entropy. The natural question is, if we generalize this situation to Tsallis or Rényi entropies, would it define good coherence measures? In other words, we define a difference between Tsallis entropies of the original state and the incoherent state closest to it when measured by Tsallis relative entropy. Taking Rényi entropy instead of the Tsallis entropy, leads to the well-known distance-based Rényi coherence, which means this expression defined a good coherence measure. Interestingly, we show that Tsallis entropy does not generate even a genuine coherence monotone, unless it is under a very restrictive class of operations. Additionally, we provide continuity estimates for both Tsallis and Rényi coherence expressions. Furthermore, we present two coherence measures based on the closest incoherent state when measures by Tsallis or Rényi relative entropy.

arXiv:2208.06840Aug 14, 20220 citations

Nearly Sample-Optimal Estimators for Quantum Rényi and Tsallis Entropies

Kean Chen, Qisheng Wang

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

Some properties of Tsallis and Tsallis–Lin quantum relative entropies

Reza Asgharzadeh Jelodar, Hossein Mehri‐Dehnavi, Hamzeh Agahi

Physica A Statistical Mechanics and its ApplicationsDec 31, 20204 citations

Renyi Entropy and the Uncertainty Relations

Iwo Białynicki‐Birula

AIP conference proceedingsJan 1, 200743 citations

A bound on expectation values and variances of quantum observables via Rényi entropy and Tsallis entropy

Sujuan Zhang, Jing Li

Entropy is a key concept of quantum information theory. The entropy of a quantum system is a measure of its randomness and has many applications in quantum communication protocols, quantum coherence, and so on. In this paper, based on the Rényi entropy and Tsallis entropy, we derive the bounds of the expectation value and variance of a quantum observable. By the maximal value of Rényi entropy, we show an upper bound on the product of variance and entropy. Furthermore, we obtain the reverse uncertainty relation for the product and sum of the variances for [Formula: see text] observables respectively.Entropy is a key concept of quantum information theory. The entropy of a quantum system is a measure of its randomness and has many applications in quantum communication protocols, quantum coherence, and so on. In this paper, based on the Rényi entropy and Tsallis entropy, we derive the bounds of the expectation value and variance of a quantum observable. By the maximal value of Rényi entropy, we show an upper bound on the product of variance and entropy. Furthermore, we obtain the reverse uncertainty relation for the product and sum of the variances for [Formula: see text] observables respectively.

International Journal of Quantum InformationApr 1, 20212 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).