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

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

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.

References

Bhatia, R., Matrix Analysis (Springer-Verlag, New York, 1996).Google Scholar
Ebadian, A., Nikoufar, I., and Eshagi Gordji, M., Perspectives of matrix convex functions, Proc. Natl. Acad. Sci. 108(18) (2011), 7313–7314.CrossRefGoogle Scholar
Effros, E. G., A matrix convexity approach to some celebrated quantum inequalities, Proc. Natl. Acad. Sci. U S A. 106(4) (2009), 1006–1008.CrossRefGoogle ScholarPubMed
Effros, E. G. and Hansen, F., Non-commutative perspectives, Ann. Funct. Anal. 5(2) (2014), 74–79.CrossRefGoogle Scholar
Fujii, J. I. and Kamei, E., Relative operator entropy in noncommutative information theory, Math. Japonica. 34 (1989), 341–348.Google Scholar
Furuichi, S., Yanagi, K., and Kuriyama, K., A note on operator inequalities of Tsallis relative operator entropy, Linear Alg. Appl. 407 (2005), 19–31.CrossRefGoogle Scholar
Furuta, T., Parametric extensions of Shannon inequality and its reverse one in Hilbert space operators, Linear Algebra Appl. 381 (2004) 219–235.CrossRefGoogle Scholar
Hansen, F. and Pedersen, G., Jensen’s Inequality for Operators and Löwner’s Theorem, Math. Ann. 258 (1982), 229–241.CrossRefGoogle Scholar
Hansen, F. and Pedersen, G., Jensen’s operator inequality, Bull. London Math. Soc. 35 (2003), 553–564.CrossRefGoogle Scholar
Hiai, F., Mosonyi, M., Petz, D., and Beny, C., Quantum f-divergences and error correction, Rev. Math. Phys. 23 (2011), 691–747.CrossRefGoogle Scholar
Jencova, A. and Ruskai, M. B., A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality, Rev. Math. Phys. 22 (2010), 1099–1121.CrossRefGoogle Scholar
Kubo, F. and Ando, T., Means of positive linear operators, Math. Ann. 246 (1979-1980), 205–224.CrossRefGoogle Scholar
Lieb, E. and Ruskai, M., Proof of the strong subadditivity of quantum-mechanical entropy, With an appendix by B. Simon, J. Math. Phys. 14 (1973), 1938–1941.CrossRefGoogle Scholar
Nakamura, M. and Umegaki, H., A note on the entropy for operator algebras, Proc. Japan Acad. 37 (1967), 149–154.Google Scholar
Nikoufar, I., On operator inequalities of some relative operator entropies, Adv. Math. 259 (2014), 376–383.CrossRefGoogle Scholar
Umegaki, H., Conditional expectation in operator algebra IV (entropy and information), Kodai Math. Sem. Rep. 14 (1962), 59–85.CrossRefGoogle Scholar
Yanagi, K., Kuriyama, K., and Furuichi, S., Generalized Shannon inequalities based on Tsallis relative operator entropy, Linear Alg. Appl. 394 (2005), 109–118.CrossRefGoogle Scholar