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CONVEXITY OF PARAMETER EXTENSIONS OF SOME RELATIVE OPERATOR ENTROPIES WITH A PERSPECTIVE APPROACH

Part of: Groups and algebras in quantum theory Communication, information General theory of linear operators

Published online by Cambridge University Press:  06 June 2019

ISMAIL NIKOUFAR
ISMAIL NIKOUFAR*
Affiliation:
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran e-mails: nikoufar@pnu.ac.ir
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Abstract

In this paper, we introduce two notions of a relative operator (α, β)-entropy and a Tsallis relative operator (α, β)-entropy as two parameter extensions of the relative operator entropy and the Tsallis relative operator entropy. We apply a perspective approach to prove the joint convexity or concavity of these new notions, under certain conditions concerning α and β. Indeed, we give the parametric extensions, but in such a manner that they remain jointly convex or jointly concave.

Significance Statement. What is novel here is that we convincingly demonstrate how our techniques can be used to give simple proofs for the old and new theorems for the functions that are relevant to quantum statistics. Our proof strategy shows that the joint convexity of the perspective of some functions plays a crucial role to give simple proofs for the joint convexity (resp. concavity) of some relative operator entropies.

MSC classification

Primary: 81P45: Quantum information, communication, networks 47A63: Operator inequalities
Secondary: 94A17: Measures of information, entropy 81R15: Operator algebra methods
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 3 , September 2020 , pp. 737 - 744
Copyright
© Glasgow Mathematical Journal Trust 2019

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Footnotes

The notions introduced here were used in our published paper [15], when this paper was a draft.

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