Acta mathematica scientia,Series B ›› 2023, Vol. 43 ›› Issue (5): 2320-2332.doi: 10.1007/s10473-023-0523-7

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RIGHT MEAN FOR THE $\alpha$-z BURES-WASSERSTEIN QUANTUM DIVERGENCE*

Miran Jeong1, Jinmi Hwangm2, Sejong Kim1†   

  1. 1. Department of Mathematics, Chungbuk National University, Cheongju 28644, Korea;
    2. Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
  • Received:2022-06-29 Revised:2022-11-09 Published:2023-10-25
  • Contact: †Sejong Kim, E-mail: skim@chungbuk.ac.kr
  • About author:Miran Jeong, E-mail: jmr4006@chungbuk.ac.kr; Jinmi Hwangm, E-mail: jinmi0401@skku.edu
  • Supported by:
    The work of S. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2022R1A2C4001306). The work of J. Hwang was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2022R1I1A1A01068411).

Abstract: The optimization problem to minimize the weighted sum of $\alpha$-z Bures-Wasserstein quantum divergences to given positive definite Hermitian matrices has been solved. We call the unique minimizer the $\alpha$-z weighted right mean, which provides a new non-commutative version of generalized mean (Hölder mean). We investigate its fundamental properties, and give many interesting operator inequalities with the matrix power mean including the Cartan mean. Moreover, we verify the trace inequality with the Wasserstein mean and provide bounds for the Hadamard product of two right means.

Key words: Rényi relative entropy, Bures-Wasserstein quantum divergence, right mean, power mean, Cartan mean, Wasserstein mean

CLC Number: 

  • 81P17
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