Korean J. Math. Vol. 26 No. 3 (2018) pp.349-371
DOI: https://doi.org/10.11568/kjm.2018.26.3.349

Inequalities for quantum $f$-divergence of convex functions and matrices

Main Article Content

Silvestru Sever Dragomir

Abstract

Some inequalities for quantum $f$-divergence of matrices are obtained. It is shown that for normalised convex functions it is nonnegative. Some upper bounds for quantum $f$-divergence in terms of variational and $\chi ^{2}$-distance are provided. Applications for some classes of divergence measures such as Umegaki and Tsallis relative entropies are also given.



Article Details

References

[1] P. Cerone and S. S. Dragomir, Approximation of the integral mean divergence and f-divergence via mean results, Math. Comput. Modelling 42 (1-2) (2005), 207–219. Google Scholar

[2] P. Cerone, S. S. Dragomir and F. O ̈sterreicher, Bounds on extended f- divergences for a variety of classes, Kybernetika (Prague) 40 (6) (2004), 745– 756. Preprint, RGMIA Res. Rep. Coll. 6 (1) (2003), Article 5. [ONLINE: http://rgmia.vu.edu.au/v6n1.html]. Google Scholar

[3] I. Csisz ar, Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizit at von Markoffschen Ketten, (German) Magyar Tud. Akad. Mat. Kutat o Int. K ozl. 8 (1963), 85-108. Google Scholar

[4] S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (3) (2006), 471–476. Google Scholar

[5] S. S. Dragomir, Some inequalities for (m, M )-convex mappings and applications for the Csisz ar Ph-divergence in information theory, Math. J. Ibaraki Univ. 33 (2001), 35-50. Google Scholar

[6] S. S. Dragomir, Some inequalities for two Csisz ar divergences and applications, Mat. Bilten. 25 (2001), 73-90. Google Scholar

[7] S. S. Dragomir, An upper bound for the Csisz ar f-divergence in terms of the variational distance and applications, Panamer. Math. J. 12 (4) (2002), 43-54. Google Scholar

[8] S. S. Dragomir, Upper and lower bounds for Csisz ar f-divergence in terms of Hellinger discrimination and applications, Nonlinear Anal. Forum 7 (1) (2002), 1-13. Google Scholar

[9] S. S. Dragomir, Bounds for f-divergences under likelihood ratio constraints, Appl. Math. 48 (3) (2003), 205–223. Google Scholar

[10] S. S. Dragomir, New inequalities for Csisza r divergence and applications, Acta Math. Vietnam. 28 (2) (2003), 123-134. Google Scholar

[11] S. S. Dragomir, A generalized f-divergence for probability vectors and applications, Panamer. Math. J. 13 (4) (2003), 61–69. Google Scholar

[12] S. S. Dragomir, Some inequalities for the Csisz ar ph-divergence when ph is an L-Lipschitzian function and applications, Ital. J. Pure Appl. Math. 15 (2004), 57-76. Google Scholar

[13] S. S. Dragomir, A converse inequality for the Csisz ar Ph-divergence, Tamsui Oxf. J. Math. Sci. 20 (1) (2004), 35-53. Google Scholar

[14] S. S. Dragomir, Some general divergence measures for probability distributions, Acta Math. Hungar. 109 (4) (2005), 331–345. Google Scholar

[15] S. S. Dragomir, A refinement of Jensen’s inequality with applications for f- divergence measures, Taiwanese J. Math. 14 (1) (2010), 153–164. Google Scholar

[16] S. S. Dragomir, A generalization of f-divergence measure to convex functions defined on linear spaces, Commun. Math. Anal. 15 (2) (2013), 1–14. Google Scholar

[17] F. Hiai, Fumio and D. Petz, From quasi-entropy to various quantum information quantities, Publ. Res. Inst. Math. Sci. 48 (3) (2012), 525–542. Google Scholar

[18] F. Hiai, M. Mosonyi, D. Petz and C. B eny, Quantum f-divergences and error correction, Rev. Math. Phys. 23 (7) (2011), 691-747. Google Scholar

[19] P. Kafka, F. O ̈sterreicher and I. Vincze, On powers of f-divergence defining a distance, Studia Sci. Math. Hungar. 26(1991), 415–422. Google Scholar

[20] F. Liese and I. Vajda, Convex Statistical Distances, Teubuer – Texte zur Math- ematik, Band 95, Leipzig, 1987. Google Scholar

[21] F. O ̈sterreicher and I. Vajda, A new class of metric divergences on probability spaces and its applicability in statistics, Ann. Inst. Statist. Math. 55 (3) (2003), 639–653. Google Scholar

[22] D. Petz, Quasi-entropies for states of a von Neumann algebra, Publ. RIMS. Kyoto Univ. 21 (1985), 781–800. Google Scholar

[23] D. Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys. 23 (1986), 57–65. Google Scholar

[24] D. Petz, From quasi-entropy, Ann. Univ. Sci. Budapest. E ̈otv ̈os Sect. Math. 55 (2012), 81–92. Google Scholar

[25] D. Petz, From f-divergence to quantum quasi-entropies and their use, Entropy 12 (3) (2010), 304–325. Google Scholar

[26] M. B. Ruskai, Inequalities for traces on von Neumann algebras, Commun. Math. Phys. 26 (1972), 280—289. Google Scholar