Hiroki Suyari

Generalization of Shannon-Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy

Tsallis entropy, one-parameter generalization of Shannon entropy, has been often discussed in statistical physics as a new information measure. This new information measure has provided many satisfactory physical interpretations in nonextensive systems exhibiting chaos or fractal. We present the generalized Shannon-Khinchin axioms to nonextensive systems and prove the uniqueness theorem rigorously. Our results show that Tsallis entropy is the simplest among all nonextensive entropies. By the detailed comparisons of our axioms with the previously presented two sets of axioms, we reveal the peculiarity of pseudoadditivity as an axiom. In this correspondence, the most fundamental basis for Tsallis entropy as information measure is established in the information-theoretic framework.

A two-parameter generalization of Shannon–Khinchin axioms and the uniqueness theorem

Abstract Based on the one-parameter generalization of Shannon–Khinchin (SK) axioms presented by one of the authors, and utilizing a tree-graphical representation, we have developed for the first time a two-parameter generalization of the SK axioms in accordance with the two-parameter entropy introduced by Sharma, Taneja, and Mittal. The corresponding unique theorem is also proved. It is found that our two-parameter generalization of Shannon additivity is a natural consequence from the Leibniz product rule of the two-parameter Chakrabarti–Jagannathan difference operator.

The unique non self-referential q-Canonical distribution and the physical temperature derived from the maximum entropy principle in tsallis statistics

The maximum entropy principle in Tsallis statistics is reformulated in the mathematical framework of the q-product, which results in the unique non self-referential q-canonical distribution. As one of the applications of the present formalism, we theoretically derive the physical temperature which coincides with that already obtained in accordance with the generalized zeroth law of thermodynamics.

Multiplicative duality, q-triplet and (μ,ν,q)-relation derived from the one-to-one correspondence between the (μ,ν)-multinomial coefficient and Tsallis entropy Sq

We derive the multiplicative duality “q↔1/q” and other typical mathematical structures as the special cases of the (μ,ν,q)-relation behind Tsallis statistics by means of the (μ,ν)-multinomial coefficient. Recently the additive duality “q↔2-q” in Tsallis statistics is derived in the form of the one-to-one correspondence between the q-multinomial coefficient and Tsallis entropy. A slight generalization of this correspondence for the multiplicative duality requires the (μ,ν)-multinomial coefficient as a generalization of the q-multinomial coefficient. This combinatorial formalism provides us with the one-to-one correspondence between the (μ,ν)-multinomial coefficient and Tsallis entropy Sq, which determines a concrete relation among three parameters μ,ν and q, i.e., ν(1-μ)+1=q which is called “(μ,ν,q)-relation” in this paper. As special cases of the (μ,ν,q)-relation, the additive duality and the multiplicative duality are recovered when ν=1 and ν=q, respectively. As other special cases, when ν=2-q, a set of three parameters (μ,ν,q) is identified with the q-triplet (qsen,qrel,qstat) recently conjectured by Tsallis. Moreover, when ν=1/q, the relation 1/(1-qsen)=1/αmin-1/αmax in the multifractal singularity spectrum f(α) is recovered by means of the (μ,ν,q)-relation.

Nonextensive entropies derived from form invariance of pseudoadditivity

The form invariance of pseudoadditivity is shown to determine the structure of nonextensive entropies. Nonextensive entropy is defined as the appropriate expectation value of nonextensive information content, similar to the definition of Shannon entropy. Information content in a nonextensive system is obtained uniquely from generalized axioms by replacing the usual additivity with pseudoadditivity. The satisfaction of the form invariance of the pseudoadditivity of nonextensive entropy and its information content is found to require the normalization of nonextensive entropies. The proposed principle requires the same normalization as that derived previously [A.K. Rajagopal and S. Abe, Phys. Rev. Lett. 83, 1711 (1999)], but is simpler and establishes a basis for the systematic definition of various entropies in nonextensive systems.

On the most concise set of axioms and the uniqueness theorem for Tsallis entropy

We present the most concise set of axioms for Tsallis entropy, and rigorously prove the uniqueness theorem. This set of axioms consists of only two distinct additivities: pseudoadditivity and Shannon additivity. We then compare our axioms with the axioms presented by Santos. The peculiarity of pseudoadditivity as an axiom for Tsallis entropy is also discussed.

Mathematical structures derived from the q-product uniquely determined by tsallis entropy

For a unified description of power-law behaviors such as chaos, fractal and scale-free network, Tsallis entropy has been applied to the generalization of the traditional Boltzmann-Gibbs statistics as a fundamental information measure. Tsallis entropy Sq is an one-parameter generalization of Shannon entropy S1 in the sense that limqrarr1 Sq = Si. The generalized Boltzmann-Gibbs statistics by means of Tsallis entropy is nowadays called Tsallis statistics. The main approach in Tsallis statistics has been the maximum entropy principle, but there have been missing some fundamental mathematical formulae such as law of error, q-Stirlings formula and q-multinomial coefficient. Recently, we have succeeded in proving law of error in Tsallis statistics using the q-product uniquely determined by Tsallis entropy. Along the same lines as the proof, we present q-Stirlings formula, q-multinomial coefficient and a conjecture on the q-central limit theorem in Tsallis statistics

Characterization of quantum teleportation processes by nonlinear quantum channel and quantum mutual entropy

Abstract The quantum mutual entropy was introduced by one of the present authors in 1983 as a quantum extension of the Shannon mutual information. It has been used for several studies such as quantum information transmission in optical communication and quantum irreversible processes. In this paper, a nonlinear channel for a quantum teleportation process is rigorously constructed and the quantum mutual entropy is applied to characterize the quantum teleportation processes of Bennett et al.

Tsallis entropy as a lower bound of average description length for the q-generalized code tree

We prove that the generalized Shannon additivity determines a lower bound of average description length for the q-generalized Z3-ary code tree. To clarify our main result, at first it is shown that the original Shannon additivity determines a lower bound of average code length of a Z3-ary code tree. As its generalization, we present our main result mentioned above. The generalized Shannon additivity is one of the generalized Shannon-Khinchin axioms for Tsallis entropy, i.e. one-parameter generalization of Shannon entropy. This reveals that Tsallis entropy is a lower bound of average description length for the q-generalized Z3-ary code tree.

Tsallis differential entropy and divergences derived from the generalized Shannon-Khinchin axioms

In discrete systems, Shannon entropy is well known to be characterized by the Shannon-Khinchin axioms. Recently, this set of axioms was generalized for Tsallis entropy, one-parameter generalization of Shannon entropy. In continuos systems, Shannon differential entropy has been introduced as a natural extension of the above Shannon entropy without using an axiomatic approach. We derive the generalized entropy function as a solution of the functional equation determined by the generalized Shannon additivity, one of the most important axiom of the generalized Shannon-Khinchin axioms for Tsallis entropy. This generalized entropy function naturally introduces Tsallis differential entropy and two Tsallis divergences. In particular, one (Csiszár type) of the divergences has almost the same form as the α-divergence in information geometry and the other the Bregman type divergence. Our results reveal that the generalized Shannon additivity representing a branch structure of a rooted tree plays an essential role in the determination of these entropies.