Generalized Shannon inequalities based on Tsallis relative operator entropy

Yanagi, Kenjiro; Kuriyama, Ken; Furuichi, Shigeru

doi:10.1016/j.laa.2004.06.025

Cited by 57 publications

(28 citation statements)

References 6 publications

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“…This paper continues the study of Tsallis relative operator entropy started in [15]. Let B (H) be the C * -algebra of all (bounded linear) operators on a Hilbert space H. The order relation A ≤ B for A, B ∈ B (H) means that both A and B are self-adjoint and B − A is positive.…”

Section: Introductionmentioning

confidence: 67%

A complementary inequality to the information monotonicity for Tsallis relative operator entropy

Moradi

Furuichi

2018

Linear and Multilinear Algebra

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We establish a reverse inequality for Tsallis relative operator entropy involving a positive linear map. In addition, we present converse of Ando's inequality, for each parameter.We give examples to compare our results with the known results by Furuta and Seo. In particular, we establish an extension and a reverse of the Löwner-Heinz inequality under certain condition. Some interesting consequences of inner product spaces and norm inequalities are also presented.

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Section: Introductionmentioning

confidence: 67%

A complementary inequality to the information monotonicity for Tsallis relative operator entropy

Moradi

Furuichi

2018

Linear and Multilinear Algebra

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“…In our previous papers [17,18], we introduced the Tsallis relative operator entropy T q (ρ|σ) as a parametric extension of the relative operator entropy S(ρ|σ) such as…”

Section: Quantum Tsallis Relative Entropy and Its Propertiesmentioning

confidence: 99%

Fundamental properties of Tsallis relative entropy

Furuichi

Yanagi

Kuriyama

2004

Journal of Mathematical Physics

150

146

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Abstract. 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. The generalized Tsallis relative entropy is defined and its subadditivity in the special case is shown by its joint convexity. As a byproduct, the superadditivity of the quantum Tsallis entropy for the independent systems in the case of 0 ≤ q < 1 is obtained. Moreover, the generalized Peierls-Bogoliubov inequality is also proven.

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“…Very recently, Tsallis relative operator entropy T p (A|B) in Yanagi-Kuriyama-Furuichi [12] is defined by…”

Section: S(a B) = Tr[a(log a − Log B)]mentioning

confidence: 99%

Reverse inequalities involving two relative operator entropies and two relative entropies

Furuta

2005

Linear Algebra and its Applications

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We shall discuss relation among Tsallis relative operator entropy T p (A|B), the relative operator entropyŜ(A|B) by J.I. Fujii-Kamei, the Tsallis relative entropy D p (A B) by Furuichi-Yanagi-Kuriyama and the Umegaki relative entropy S (A, B). We show the following result: Let A and B be strictly positive definite matrices such that M 1 I A m 1 I > 0 and M 2 I B m 2 I > 0. Put h = M 1 M 2 m 1 m 2 > 1 and p ∈ (0, 1]. Then the following inequalities hold:

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Generalized Shannon inequalities based on Tsallis relative operator entropy

Cited by 57 publications

References 6 publications

A complementary inequality to the information monotonicity for Tsallis relative operator entropy

A complementary inequality to the information monotonicity for Tsallis relative operator entropy

Fundamental properties of Tsallis relative entropy

Reverse inequalities involving two relative operator entropies and two relative entropies

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