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

doi:10.1007/s10473-023-0523-7

数学物理学报(英文版) ›› 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 Kim^1†

1. Department of Mathematics, Chungbuk National University, Cheongju 28644, Korea;
2. Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea

收稿日期:2022-06-29 修回日期:2022-11-09 发布日期:2023-10-25

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

Miran Jeong¹, Jinmi Hwangm², Sejong Kim^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).

摘要/Abstract

摘要： 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.

关键词: Rényi relative entropy, Bures-Wasserstein quantum divergence, right mean, power mean, Cartan mean, Wasserstein mean

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

中图分类号:

81P17

Miran Jeong, Jinmi Hwangm, Sejong Kim. RIGHT MEAN FOR THE $\alpha$-z BURES-WASSERSTEIN QUANTUM DIVERGENCE^*[J]. 数学物理学报(英文版), 2023, 43(5): 2320-2332.

Miran Jeong, Jinmi Hwangm, Sejong Kim. RIGHT MEAN FOR THE $\alpha$-z BURES-WASSERSTEIN QUANTUM DIVERGENCE^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2320-2332.

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

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

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