Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

INEQUALITIES FOR QUANTUM f-DIVERGENCE OF CONVEX FUNCTIONS AND MATRICES

  • Received : 2018.02.15
  • Accepted : 2018.08.14
  • Published : 2018.09.30
Download PDF
⟨ Previous Next ⟩

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.

Keywords

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. https://doi.org/10.1016/j.mcm.2004.02.044
  2. P. Cerone, S. S. Dragomir and F. Osterreicher, 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].
  3. I. Csiszar, Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten, (German) Magyar Tud. Akad. Mat. Kutato Int. Kozl. 8 (1963), 85-108.
  4. S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (3) (2006), 471-476. https://doi.org/10.1017/S000497270004051X
  5. S. S. Dragomir, Some inequalities for (m, M)-convex mappings and applications for the Csiszar $\Phi$-divergence in information theory, Math. J. Ibaraki Univ. 33 (2001), 35-50. https://doi.org/10.5036/mjiu.33.35
  6. S. S. Dragomir, Some inequalities for two Csiszar divergences and applications, Mat. Bilten. 25 (2001), 73-90.
  7. S. S. Dragomir, An upper bound for the Csiszar f-divergence in terms of the variational distance and applications, Panamer. Math. J. 12 (4) (2002), 43-54.
  8. S. S. Dragomir, Upper and lower bounds for Csiszar f-divergence in terms of Hellinger discrimination and applications, Nonlinear Anal. Forum 7 (1) (2002), 1-13.
  9. S. S. Dragomir, Bounds for f-divergences under likelihood ratio constraints, Appl. Math. 48 (3) (2003), 205-223. https://doi.org/10.1023/A:1026054413327
  10. S. S. Dragomir, New inequalities for Csiszar divergence and applications, Acta Math. Vietnam. 28 (2) (2003), 123-134.
  11. S. S. Dragomir, A generalized f-divergence for probability vectors and applications, Panamer. Math. J. 13 (4) (2003), 61-69.
  12. S. S. Dragomir, Some inequalities for the Csiszar $\varphi$-divergence when $\varphi$ is an L-Lipschitzian function and applications, Ital. J. Pure Appl. Math. 15 (2004), 57-76.
  13. S. S. Dragomir, A converse inequality for the Csiszar $\Phi$-divergence, Tamsui Oxf. J. Math. Sci. 20 (1) (2004), 35-53.
  14. S. S. Dragomir, Some general divergence measures for probability distributions, Acta Math. Hungar. 109 (4) (2005), 331-345. https://doi.org/10.1007/s10474-005-0251-6
  15. S. S. Dragomir, A refinement of Jensen's inequality with applications for f-divergence measures, Taiwanese J. Math. 14 (1) (2010), 153-164. https://doi.org/10.11650/twjm/1500405733
  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.
  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. https://doi.org/10.2977/PRIMS/79
  18. F. Hiai, M. Mosonyi, D. Petz and C. Beny, Quantum f-divergences and error correction, Rev. Math. Phys. 23 (7) (2011), 691-747. https://doi.org/10.1142/S0129055X11004412
  19. P. Kafka, F. Osterreicher and I. Vincze, On powers of f-divergence defining a distance, Studia Sci. Math. Hungar. 26(1991), 415-422.
  20. F. Liese and I. Vajda, Convex Statistical Distances, Teubuer - Texte zur Mathematik, Band 95, Leipzig, 1987.
  21. F. Osterreicher 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. https://doi.org/10.1007/BF02517812
  22. D. Petz, Quasi-entropies for states of a von Neumann algebra, Publ. RIMS. Kyoto Univ. 21 (1985), 781-800.
  23. D. Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys. 23 (1986), 57-65. https://doi.org/10.1016/0034-4877(86)90067-4
  24. D. Petz, From quasi-entropy, Ann. Univ. Sci. Budapest. Eotvos Sect. Math. 55 (2012), 81-92.
  25. D. Petz, From f-divergence to quantum quasi-entropies and their use, Entropy 12 (3) (2010), 304-325. https://doi.org/10.3390/e12030304
  26. M. B. Ruskai, Inequalities for traces on von Neumann algebras, Commun. Math. Phys. 26 (1972), 280-289. https://doi.org/10.1007/BF01645523