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

Nonadditive information measure and quantum entanglement in a class of mixed states of an Nn system

Sumiyoshi Abe

Generalizing Khinchin's classical axiomatic foundation, a basis is developed for nonadditive information theory with the Tsallis entropy indexed by q. The classical nonadditive conditional entropy is introduced and then translated into quantum theory. To examine if this theory has points superior to the ordinary additive information theory with the von Neumann entropy corresponding to the limit $\stackrel{\ensuremath{\rightarrow}}{q}1,$ separability of a one-parameter family of the Werner-Popescu states of the ${N}^{n}$ system (i.e., the n-partite N-level system) is discussed. The nonadditive information theory with $q>1$ is shown to yield a limitation on separability that is stronger than the one derived from the additive theory. How the strongest limitation can be obtained in the limit $\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty}$ is also shown.Generalizing Khinchin's classical axiomatic foundation, a basis is developed for nonadditive information theory with the Tsallis entropy indexed by q. The classical nonadditive conditional entropy is introduced and then translated into quantum theory. To examine if this theory has points superior to the ordinary additive information theory with the von Neumann entropy corresponding to the limit $\stackrel{\ensuremath{\rightarrow}}{q}1,$ separability of a one-parameter family of the Werner-Popescu states of the ${N}^{n}$ system (i.e., the n-partite N-level system) is discussed. The nonadditive information theory with $q>1$ is shown to yield a limitation on separability that is stronger than the one derived from the additive theory. How the strongest limitation can be obtained in the limit $\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty}$ is also shown.

arXiv:quant-ph/0104135Physical Review AMay 8, 200238 citations

Conditional entropy in algebraic statistical physics

V. P. Belavkin, P. Staszewski

Reports on Mathematical PhysicsDec 1, 19847 citations

{N}^{n}

system (i.e., the n-partite N-level system) is discussed. The nonadditive information theory with

q>1

is shown to yield a limitation on separability that is stronger than the one derived from the additive theory. How the strongest limitation can be obtained in the limit $\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty}$ is also shown.