A generalized Fannes' inequality

茂古市, 研二郎柳, 憲栗山

Published Jan 1, 2007• 0 citations

Abstract

We axiomatically characterize the Tsallis entropy of a finite quantum system.In addition, we derive a continuity property of Tsallis entropy.This gives a generalization of the Fannes' inequality.

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A generalized Fannes' inequality

茂古市, 研二郎柳, 憲栗山

Published Jan 1, 2007• 0 citations

OpenAlex

Abstract

We axiomatically characterize the Tsallis entropy of a finite quantum system.In addition, we derive a continuity property of Tsallis entropy.This gives a generalization of the Fannes' inequality.

Related papers

A generalized Fannes inequality

Shigeru Furuichi, Kenjiro Yanagi, Ken Kuriyama

We axiomatically characterize the Tsallis entropy of a finite quantum system. In addition, we derive a continuity property of Tsallis entropy. This gives a generalization of the Fannes inequality.

arXiv:1001.1390Journal of Inequalities in Pure & Applied MathematicsJan 9, 201011 citations

A review of the mathematical properties of the Tsallis entropies

Shigeru Furuichi, Alexander M. Korsunsky

We review the mathematical properties of the generalized entropies, based on our previous papers. This review is an extended version of our article [9]. The Tsallis entropy defined for a quantum state is uniquely derived from three axioms. A generalized Fannes’ inequality is shown. The maximum entropy principle in Tsallis statistics are revisited as an application of the Tsallis relative entropy defined for nonnegative matrices in the framework of matrix analysis. A variational expression for Tsallis relative entropy is derived and some related inequalities are studied. Finally, an application to qunatum information theory is discussed.We review the mathematical properties of the generalized entropies, based on our previous papers. This review is an extended version of our article [9]. The Tsallis entropy defined for a quantum state is uniquely derived from three axioms. A generalized Fannes’ inequality is shown. The maximum entropy principle in Tsallis statistics are revisited as an application of the Tsallis relative entropy defined for nonnegative matrices in the framework of matrix analysis. A variational expression for Tsallis relative entropy is derived and some related inequalities are studied. Finally, an application to qunatum information theory is discussed.

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math-phCross-listedarXiv:2512.01745Journal of Mathematical PhysicsDec 2, 20250 citations

For the sandwiched Rényi entropy the conditional entropy can be defined two ways:

\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{H}^\uparrow_α(A|B)_ρ

. In the limiting case,

α=1

, both definitions consolidate into conditional entropy

H(A|B)=S(AB)-S(B)

. The continuity inequality for conditional entropy

H(A|B)

, called the Alicki-Fannes-Winter (AWF) inequality, shows that if the states are close in trace-distance, then the conditional entropies are also close. Having the AWF inequality for conditional entropy, we show that the channel entropy defined through the relative entropy is continuous with respect to the diamond-distance between channels. Inspired by this, similar continuity inequalities for the Rényi conditional entropy

\widetilde{H}^\uparrow_α

were obtained in the work [A. Marwah and F. Dupuis, J. Math. Phys. 63, 052201 (2022)]. We provide continuity bounds for the sandwiched Rényi and Tsallis conditional entropies

\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{T}^\downarrow_α(A|B)_ρ

for states with the same marginal on the conditioning system. Similar to the previous bounds, our bound depends only on the dimension of the conditioning system. We apply this result to prove continuity of the channel entropy for Rényi and Tsallis channel entropies defined through the sandwiched Rényi and Tsallis relative entropies.