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Relative Operator Entropy

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Operator and Norm Inequalities and Related Topics

Part of the book series: Trends in Mathematics ((TM))

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Abstract

The relative operator entropy S(A|B) is an operator version of (the minis of) the Kullback-Leibler divergence in the information theory. It is introduced an extension, called solidarities A s B, of the Kubo-Ando operator means A m B and moreover it is a tangent vector of the path of geometric means A # t B. So we discuss mean-like properties and geometric ones in the manifold of the positive invertible operators. In fact, this path A # t B is a geodesic and S(A|B) is the initial tangent vector in this manifold with the principal fiber bundle, say the CPR geometry. The former defines the multivariate power mean and the latter the Karcher mean. Related to the quantum information theory, we discuss the Tsallis operator entropy and its trace as the secant vector between A and A # t B.

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Acknowledgements

The authors are partially supported by JSPS KAKENHI Grant Number JP19K03542.

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Authors and Affiliations

  1. Department of Educational Collaboration (Sciences; Mathematics and Information), Osaka Kyoiku University, Kashiwara, Osaka, Japan

    Jun Ichi Fujii

  2. Department of Mathematics Education, Osaka Kyoiku University, Kashiwara, Osaka, Japan

    Yuki Seo

Authors
  1. Jun Ichi Fujii
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  2. Yuki Seo
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Correspondence to Yuki Seo .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, OH, USA

    Richard M. Aron

  2. Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

    Mohammad Sal Moslehian

  3. Department of Mathematics, New York University Abu Dhabi, Abu Dhabi, United Arab Emirates

    Ilya M. Spitkovsky

  4. Department of Mathematics, Drexel University, Philadelphia, PA, USA

    Hugo J. Woerdeman

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Fujii, J.I., Seo, Y. (2022). Relative Operator Entropy. In: Aron, R.M., Moslehian, M.S., Spitkovsky, I.M., Woerdeman, H.J. (eds) Operator and Norm Inequalities and Related Topics. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-02104-6_3

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