Shigeru Furuichi

Information theoretical properties of Tsallis entropies

A chain rule and a subadditivity for the entropy of type β, which is one of the nonadditive entropies, were derived by Daroczy. In this paper, we study the further relations among Tsallis type entropies which are typical nonadditive entropies. The chain rule is generalized by showing it for Tsallis relative entropy and the nonadditive entropy. We show some inequalities related to Tsallis entropies, especially the strong subadditivity for Tsallis type entropies and the subadditivity for the nonadditive entropies. The subadditivity and the strong subadditivity naturally lead to define Tsallis mutual entropy and Tsallis conditional mutual entropy, respectively, and then we show again chain rules for Tsallis mutual entropies. We give properties of entropic distances in terms of Tsallis entropies. Finally we show parametrically extended results based on information theory.

Fundamental properties of Tsallis relative entropy

Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz. The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible. The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity. Moreover, the generalized Peierls–Bogoliubov inequality is also proven.

A generalized skew information and uncertainty relation

A generalized skew information is defined and a generalized uncertainty relation is established with the help of a trace inequality which was recently proven by Fujii. In addition, we prove the trace inequality conjectured by Luo and Zhang. Finally, we point out that Theorem 1 in S. Luo and Q. Zhang, IEEE Trans. Inf. Theory, vol. 50, pp. 1778-1782, no. 8, Aug. 2004 is incorrect in general, by giving a simple counter-example.

Entanglement in a squeezed two-level atom

In a previous paper, we adopted the method using quantum mutual entropy to measure the degree of entanglement in the time development of the Jaynes–Cummings model. In this paper, we formulate the entanglement in the time development of the Jaynes–Cummings model with squeezed states, and then show that the entanglement can be controlled by means of squeezing.

On uniqueness Theorems for Tsallis entropy and Tsallis relative entropy

The uniqueness theorem for Tsallis entropy was presented in H. Suyari, IEEE Trans. Inf. Theory, vol. 50, pp. 1783-1787, Aug. 2004 by introducing the generalized Shannon-Khinchin axiom. In the present correspondence, this result is generalized and simplified as follows: Generalization : The uniqueness theorem for Tsallis relative entropy is shown by means of the generalized Hobsons axiom. Simplification: The uniqueness theorem for Tsallis entropy is shown by means of the generalized Faddeevs axiom

ALTERNATIVE REVERSE INEQUALITIES FOR YOUNG'S INEQUALITY

Two reverse inequalities for Youngs inequality were shown by M. Tominaga, using Specht ratio. In this short paper, we show alternative reverse inequalities for Youngs inequality without using Specht ratio.

Entanglement Degree in the Time Development of the Jaynes–Cummings Model

Recently, it has been become known that a quantum entangled state plays an important role in fields of quantum information theory, such as quantum teleportation and quantum computation. Research on quantifying entangled states has been carried out using several measures. In this Letter, we will adopt this method using quantum mutual entropy to measure the degree of entanglement in the time development of the Jaynes–Cummings model.

Entanglement degree of a nonlinear multiphoton Jaynes-Cummings model

We extend our earlier investigations (Furuichi S and Abdel-Aty M 2001 J. Phys. A: Math. Gen. 34 6851) on the entanglement degree of a two-level atom to include any forms of nonlinearities of both the field and the intensity-dependent atom-field coupling. We present a derivation of the unitary operator within the frame of the dressed state approach, by means of which we identify and numerically demonstrate the region of parameters where significantly large entanglement can be obtained. The influences of the nonlinearity, Stark shifts and detuning on the degree of entanglement are examined. It is shown that features of the degree of entanglement are influenced significantly by the kinds of nonlinearity of the single-mode field. The model presented in this paper can be regarded as a generalization of the Jaynes-Cummings model.

On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics

We give a new proof of the theorems on the maximum entropy principle in Tsallis statistics. That is, we show that the q-canonical distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the q-expectation value, and the q-Gaussian distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the q-variance, as applications of the non-negativity of the Tsallis relative entropy, without using the Lagrange multipliers method. In addition, we define a q-Fisher information and then prove a q-Cramer–Rao inequality that the q-Gaussian distribution with special q-variances attains the minimum value of the q-Fisher information.

A matrix trace inequality and its application

In this short paper, we give a complete and affirmative answer to a conjecture on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson inequality for positive semidefinite matrices.