Ken Kuriyama

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.

Uncertainty relations for generalized metric adjusted skew information and generalized metric adjusted correlation measure

AbstractIn this paper, we give a Heisenberg-type or a Schrödinger-type uncertainty relation for generalized metric adjusted skew information or generalized metric adjusted correlation measure. These results generalize the previous result of Furuichi and Yanagi (J. Math. Anal. Appl. 388:1147-1156, 2012).AMSPrimary: 15A45, 47A63; secondary: 94A17

Smoothness and Monotone Decreasingness of the Solution to the BCS-Bogoliubov Gap Equation for Superconductivity

We show the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation for superconductivity. Here the temperature belongs to the closed interval

Remarks on concavity of the auxiliary function appearing in quantum reliability function

[0,\, \tau]

On bounds for symmetric divergence measures

with

Generalized Wigner-Yanase skew information and generalized fisher information

\tau>0

A Note on Sesquilinear Forms and the Generalized Uncertainty Relations in *-Algebras

nearly equal to half of the transition temperature. We show that the solution is continuous with respect to both the temperature and the energy, and that the solution is Lipschitz continuous and monotone decreasing with respect to the temperature. Moreover, we show that the solution is partially differentiable with respect to the temperature twice and the second-order partial derivative is continuous with respect to both the temperature and the energy, or that the solution is approximated by such a smooth function.

A note on operator inequalities of Tsallis relative operator entropy

Concavity of the auxiliary function which appears in the random coding exponent as the lower bound of the quantum reliability function for general quantum states is proved for 0/spl les/s/spl les/1 in two dimensional case.

Generalized Shannon inequalities based on Tsallis relative operator entropy

In the paper [1], tight bounds for symmetric divergence measures applying the results established by G.L.Gilardoni. In this article, we report on two kinds of extensions for the Sason’s results, namely a classical q-extension and a non-commutative(quantum) extension. Especially, we improve Sason’s bound of the summation of the absolute value for the difference between two probability distributions, applying the parameter q of Tsallis entropy, under a certain assumption.In the paper [1], tight bounds for symmetric divergence measures applying the results established by G.L.Gilardoni. In this article, we report on two kinds of extensions for the Sason’s results, namely a classical q-extension and a non-commutative(quantum) extension. Especially, we improve Sason’s bound of the summation of the absolute value for the difference between two probability distributions, applying the parameter q of Tsallis entropy, under a certain assumption.