Nonadditive classical information theory is developed in the axiomatic framework and then translated into quantum theory. The nonadditive conditional entropy associated with the Tsallis entropy indexed by q is given in accordance with the formalism of nonextensive statistical mechanics. The theory is applied to the problems of quantum entanglement and separability of the Werner-Popescu-type mixed state of a multipartite system, in order to examine if it has any points superior to the additive theory with the von Neumann entropy realized in the limit q → 1. It is shown that the nonadditive theory can lead to the necessary and sufficient condition for separability of the Werner-Popescu-type state, whereas the von Neumann theory can give only a much weaker condition…. Tsallis' nonextensive generalization of Boltzmann-Gibbs statistical mechanics [3, 15, 16] and its success in describing behaviors of a large class of complex systems naturally lead to the question of whether information theory can also admit an analogous generalization. If the answer is affirmative, then that will be of particular importance in connection with the problem of quantum entanglement and quantum theory of measurement [6, 8], in which necessities of a nonadditive information measure and an information content are suggested. One should also remember that there exists a conceptual similarity between a complex system and an entangled quantum system. In these systems, a "part" is indivisibly connected with the rest. An external operation on any part drastically influences the whole system, in general. Thus, the traditional reductionistic approach to an understanding of the nature of such a system may not work efficiently. In this chapter, we report a recent development in nonadditive quantum information theory based on the Tsallis entropy indexed by q [15] and its associated nonadditive conditional entropy [1]. This theory includes the ordinary additive theory with the von Neumann entropy in a special limiting case: q → To see if it has points superior to the additive theory, we apply it to the problems of separability and quantum entanglement.Nonadditive classical information theory is developed in the axiomatic framework and then translated into quantum theory. The nonadditive conditional entropy associated with the Tsallis entropy indexed by q is given in accordance with the formalism of nonextensive statistical mechanics. The theory is applied to the problems of quantum entanglement and separability of the Werner-Popescu-type mixed state of a multipartite system, in order to examine if it has any points superior to the additive theory with the von Neumann entropy realized in the limit q → 1. It is shown that the nonadditive theory can lead to the necessary and sufficient condition for separability of the Werner-Popescu-type state, whereas the von Neumann theory can give only a much weaker condition…. Tsallis' nonextensive generalization of Boltzmann-Gibbs statistical mechanics [3, 15, 16] and its success in describing behaviors of a large class of complex systems naturally lead to the question of whether information theory can also admit an analogous generalization. If the answer is affirmative, then that will be of particular importance in connection with the problem of quantum entanglement and quantum theory of measurement [6, 8], in which necessities of a nonadditive information measure and an information content are suggested. One should also remember that there exists a conceptual similarity between a complex system and an entangled quantum system. In these systems, a "part" is indivisibly connected with the rest. An external operation on any part drastically influences the whole system, in general. Thus, the traditional reductionistic approach to an understanding of the nature of such a system may not work efficiently. In this chapter, we report a recent development in nonadditive quantum information theory based on the Tsallis entropy indexed by q [15] and its associated nonadditive conditional entropy [1]. This theory includes the ordinary additive theory with the von Neumann entropy in a special limiting case: q → To see if it has points superior to the additive theory, we apply it to the problems of separability and quantum entanglement.

Oxford University Press eBooksApr 15, 20041 citation

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

On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State

Lu Wei

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

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 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.