The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy.

arXiv:1807.02212EntropyMay 27, 201919 citations

Polygamy of multiparty q -expected quantum entanglement

Jeong San Kim

We characterize the polygamy nature of quantum entanglement in multiparty systems in terms of $q$ -expectation value for the full range of $q\ensuremath{\ge}1$. By investigating some properties of generalized quantum correlations in terms of $q$ -expectation value and Tsallis $q$ -entropy, we establish a class of polygamy inequalities of multiparty quantum entanglement in arbitrary dimensions based on $q$ -expected entanglement measure. As Tsallis $q$ -entropy is reduced to von Neumann entropy, and $q$ -expectation value becomes the ordinary expectation value when $q$ tends to 1, our results encapsulate previous results of polygamy inequalities based on von Neumann entropy as special cases.We characterize the polygamy nature of quantum entanglement in multiparty systems in terms of $q$-expectation value for the full range of $q\ensuremath{\ge}1$. By investigating some properties of generalized quantum correlations in terms of $q$-expectation value and Tsallis $q$-entropy, we establish a class of polygamy inequalities of multiparty quantum entanglement in arbitrary dimensions based on $q$-expected entanglement measure. As Tsallis $q$-entropy is reduced to von Neumann entropy, and $q$-expectation value becomes the ordinary expectation value when $q$ tends to 1, our results encapsulate previous results of polygamy inequalities based on von Neumann entropy as special cases.

arXiv:1909.10346Physical Review ADec 26, 20194 citations

Computational hardness of estimating quantum entropies via binary entropy bounds

Yupan Liu

We investigate the computational hardness of estimating the quantum $α$ -Rényi entropy ${\rm S}^{\tt R}_α(ρ) = \frac{\ln {\rm Tr}(ρ^α)}{1-α}$ and the quantum $q$ -Tsallis entropy ${\rm S}^{\tt T}_q(ρ) = \frac{1-{\rm Tr}(ρ^q)}{q-1}$ , both of which converge to the von Neumann entropy as the order approaches $1$ . The promise problems Quantum $α$ -Rényi Entropy Approximation (RényiQEA $_α$ ) and Quantum $q$ -Tsallis Entropy Approximation (TsallisQEA $_q$ ) ask whether $ {\rm S}^ {\tt R}_α(ρ)$ or ${\rm S}^{\tt T}_q(ρ)$ , is at least $τ_1$ or at most $τ_2$ , where $τ_1 - τ_2$ is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order $1$ ) and some cases of the quantum $q$ -Tsallis entropy, while existing approaches do not readily extend to other orders. We establish that for all positive real $α$ and $q$ , and also for $α=\infty$ , the rank- $2$ variants Rank2RényiQEA $_α$ and Rank2TsallisQEA $_q$ are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), as well as the one derived from O'Donnell and Wright (STOC 2016), our results imply: - For all real orders $α> 0$ or $α=\infty$ , and for all real orders $0 < q \leq 1$ , LowRankRényiQEA $_α$ and LowRankTsallisQEA $_q$ are BQP-complete, where both are restricted versions of RényiQEA $_α$ and TsallisQEA $_q$ with $ρ$ of polynomial rank. - For all real order $q>1$ , TsallisQEA $_q$ is BQP-complete. Our hardness results stem from reductions based on new inequalities relating the $α$ -Rényi or $q$ -Tsallis binary entropies of different orders. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.We investigate the computational hardness of estimating the quantum $α$-Rényi entropy ${\rm S}^{\tt R}_α(ρ) = \frac{\ln {\rm Tr}(ρ^α)}{1-α}$ and the quantum $q$-Tsallis entropy ${\rm S}^{\tt T}_q(ρ) = \frac{1-{\rm Tr}(ρ^q)}{q-1}$, both of which converge to the von Neumann entropy as the order approaches $1$. The promise problems Quantum $α$-Rényi Entropy Approximation (RényiQEA$_α$) and Quantum $q$-Tsallis Entropy Approximation (TsallisQEA$_q$) ask whether $ {\rm S}^ {\tt R}_α(ρ)$ or ${\rm S}^{\tt T}_q(ρ)$, is at least $τ_1$ or at most $τ_2$, where $τ_1 - τ_2$ is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order $1$) and some cases of the quantum $q$-Tsallis entropy, while existing approaches do not readily extend to other orders. We establish that for all positive real $α$ and $q$, and also for $α=\infty$, the rank-$2$ variants Rank2RényiQEA$_α$ and Rank2TsallisQEA$_q$ are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), as well as the one derived from O'Donnell and Wright (STOC 2016), our results imply: - For all real orders $α> 0$ or $α=\infty$, and for all real orders $0 < q \leq 1$, LowRankRényiQEA$_α$ and LowRankTsallisQEA$_q$ are BQP-complete, where both are restricted versions of RényiQEA$_α$ and TsallisQEA$_q$ with $ρ$ of polynomial rank. - For all real order $q>1$, TsallisQEA$_q$ is BQP-complete. Our hardness results stem from reductions based on new inequalities relating the $α$-Rényi or $q$-Tsallis binary entropies of different orders. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.

quant-pharXiv:2601.03734Leibniz international proceedings in informaticsJan 8, 20260 citations

We characterize the polygamy nature of quantum entanglement in multiparty systems in terms of

q

-expectation value for the full range of $q\ensuremath{\ge}1$. By investigating some properties of generalized quantum correlations in terms of

q

-expectation value and Tsallis

q

-entropy, we establish a class of polygamy inequalities of multiparty quantum entanglement in arbitrary dimensions based on

q

-expected entanglement measure. As Tsallis

q

-entropy is reduced to von Neumann entropy, and

q

-expectation value becomes the ordinary expectation value when

q

tends to 1, our results encapsulate previous results of polygamy inequalities based on von Neumann entropy as special cases.

We investigate the computational hardness of estimating the quantum

α

-Rényi entropy

{\rm S}^{\tt R}_α(ρ) = \frac{\ln {\rm Tr}(ρ^α)}{1-α}

and the quantum

q

-Tsallis entropy

{\rm S}^{\tt T}_q(ρ) = \frac{1-{\rm Tr}(ρ^q)}{q-1}

, both of which converge to the von Neumann entropy as the order approaches

1

. The promise problems Quantum

α

-Rényi Entropy Approximation (RényiQEA

_α

) and Quantum

q

-Tsallis Entropy Approximation (TsallisQEA

_q

) ask whether $ {\rm S}^ {\tt R}_α(ρ)$ or

{\rm S}^{\tt T}_q(ρ)

, is at least

τ_1

or at most

τ_2

, where

τ_1 - τ_2

is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order

1

) and some cases of the quantum

q

-Tsallis entropy, while existing approaches do not readily extend to other orders. We establish that for all positive real

α

and

q

, and also for

α=\infty

, the rank-

2

variants Rank2RényiQEA

_α

and Rank2TsallisQEA

_q

are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), as well as the one derived from O'Donnell and Wright (STOC 2016), our results imply: - For all real orders

α> 0

α=\infty

, and for all real orders

0 < q \leq 1

, LowRankRényiQEA

_α

and LowRankTsallisQEA

_q

are BQP-complete, where both are restricted versions of RényiQEA

_α

and TsallisQEA

_q

with

ρ

of polynomial rank. - For all real order

q>1

, TsallisQEA

_q

is BQP-complete. Our hardness results stem from reductions based on new inequalities relating the

α

-Rényi or

q

-Tsallis binary entropies of different orders. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.