Tsallis relative operator entropy properties with some weighted metrics

Document Type : Research Article

Authors

1 LAGA-Laboratory, Science Faculty, Ibn Tofail University, Kenitra, Morocco

2 Department of Mathematics, CRMEF-RSK, EREAM Team, LaREAMI-Lab, Kenitra, Morocco

Abstract

The present work attempts to provide some properties for Tsallis relative operator entropy $T_p(A | B)$, acting on positive definite matrices, with respect to weighted Hellinger and Alpha Procrustes distances. Many localizations of this operator have been determined. In particular, some estimations of the distances between $T_p(A | B)$ and some standard matrix means are outlined.

Keywords

Main Subjects

References

[1] S. Abe, Monotone decrease of the quantum nonadditive divergence by projective measurements, Phy. Letters A 312 (2003) 336--338.
[2] R. Bhatia, T. Jain and Y. Lim, On the Bures-Wasserstein distance between positive definite matrices, Expos. Math. 37 (2019), no. 2, 165--191.
[3] M. Chergui, A. El Hilali and B. El Wahbi, On estimating some distances involving operator entropies via Riemannian metric, Khayyam J. Math. 8 (2022), no. 1, 94--101.
[4] T. H. Dinh, R. Dumitru and J. A. Franco, Some geometric properties of matrix means with respect to different metrics, Positivity, (2020), no. 5, 1419--1434.
[5] T. H. Dinh, C. T. Le and B. K. Vo, Weighted Hellinger distance and in-betweenness property, Math. Inequal. Appl. 24 (2021), no. 1, 157--165.
[6] S. S. Dragomir, Inequalities for relative operator entropy in terms of Tsallis entropy, Asian-Eur. J. Math. 12 (2019), 4, 1950063, 22 pp.
[7] F. Hiai and D. Petz, Matrix Monotone Functions and Convexity, In: Introduction to Matrix Analysis and Applications, Springer, 2014.
[8] A. El Hilali, B. El Wahbi and M. Chergui, A geometric study of relative operator entropies, Nonlinear Dyn. Syst. Theory 22 (2022), no. 1, 46--57.
[9] A. El Hilali, M. Chergui and B. El Wahbi, On Some Tsallis relative operator entropy properties related to Hellinger metrics, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 48 (2022), no. 2, 259--273.
[10] A. E. Hilali, M. Chergui, B. E. Wahbi and F. Ayoub, A Study of the Relative Operator Entropy via Some Matrix Versions of the Hellinger Distance, 2021 4th International Conference on Advanced Communication Technologies and Networking (CommNet), (2021), 1-6.
[11] J. I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon. 34 (1989), 3, 341--348.
[12] J. I. Fujii and E. Kamei, Uhlmann’s interpolational method for operator means, Math. Japon. 34 (1989), no. 4, 541--547.
[13] S. Furuichi and N. Minculete, Inequalities for relative operator entropies and operator means, Acta Math. Vietnam. 43 (2018), no. 4, 607--618.
[14] J. Havrda and F. Charvat, Quantification method of classification processes: concept of structural a-entropy, Kybernetika 3 (1967) 30--35.
[15] S. Kim, Operator entropy and fidelity associated with the geometric mean, Linear Algebra Appl. 438 (2013), no. 5, 2475--2483.
[16] H. Q. Minh, Alpha Procrustes metrics between positive definite operators: a unifying formulation for the Bures-Wasserstein and log-Euclidean/log-Hilbert-Schmidt metrics, Linear Algebra Appl. 636 (2022) 25--68.
[17] M. Nakamura and H. Umegaki, A note on the entropy for operator algebras, Proc. Japan Acad. 37 (1961) 149--154.
[18] I. Nikoufar, On operator inequalities of some relative operator entropies, Adv. Math. 259 (2014) 376--383.
[19] J. Pec̆arić, T. Furuta, J. Mićić Hot and Y. Seo, Mond-Pečarić Method in Operator Inequalities, ELEMENT, Zagreb, 2005.
[20] D. Petz, A survey of certain trace inequalities, Functional analysis and operator theory (Warsaw, 1992), 287--298, Banach Center Publ., 30, Polish Acad. Sci. Inst. Math., Warsaw, 1994.
[21] C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379--423, 623--656.
[22] D. Spehner, F. Illuminati, M. Orszag and W. Roga, Geometric Measures of Quantum Correlations with Bures and
Hellinger Distances, In: Fanchini F., Soares Pinto D., Adesso G. (eds) Lectures on General Quantum Correlations and their Applications. Quantum Science and Technology. Springer, Cham. 2017.
[23] K. Yanagi, K. Kuriyama and S. Furuichi, Generalized Shannon inequalities based on Tsallis relative operator entropy, Linear Algebra Appl. 394 (2005) 109--118.
[24] X. Yang, A matrix trace inequality, J. Math. Anal. Appl. 250 (2000), no. 1, 372--374.
[25] H. Zhou, On some trace inequalities for positive definite Hermitian matrices, J. Inequal. Appl. (2014), 2014:64.
Journal of the Iranian Mathematical Society
Volume 5, Issue 1
January 2024
Pages 55-69

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