Reverse inequalities involving two relative operator entropies and two relative entropies

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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“…There are a few other results in this direction; see, e.g., [5,11]. In the present paper, we give alternative bounds for Furuta's inequalities.…”

Section: Introductionmentioning

confidence: 78%

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

Moradi

Furuichi

2018

Linear and Multilinear Algebra

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“…The further upper bound of the right hand side was given by T.Furuta in [21], with the generalized Kantorovich constant:…”

Section: Definition 21 the Tsallis Relative Entropy Is Defined Bymentioning

confidence: 99%

Inequalities for Tsallis relative entropy and generalized skew information

Furuichi

2011

Linear and Multilinear Algebra

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Quantum entropy and skew information play important roles in quantum information science. They are defined by the trace of the positive operators so that the trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this paper, we study some properties for information quantities in quantum system through trace inequalities. Especially, we give upper bounds and lower bounds of Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum system. In addition, we compare the known bounds and the new bounds, for both upper and lower bounds, respectively. We also give an inequality for generalized skew information by introducing a generalized correlation measure.

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“…[ME, LR, L, NU]. There have been investigated the so-called entropy inequalities by some mathematicians, see [BLP,BS,F2] and references therein. A relative operator entropy of strictly positive operators A, B was introduced in the noncommutative information theory by Fujii and Kamei [FK] by…”

Section: Introductionmentioning

confidence: 99%