Nihonkai Mathematical Journal

The Quadratic Quantum $f$-Divergence of Convex Functions and Matrices

Silvestru Sever Dragomir

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper we introduce the concept of quadratic quantum $f$-divergence measure for a continuos function $f$ defined on the positive semi-axis of real numbers, the invertible matrix $T$ and matrix $V$ by $$\mathcal{S}_{f}\left( V,T\right) :=\mathrm{tr}\left[ \left\vert T^{\ast }\right\vert ^{2}f\left( \left\vert VT^{-1}\right\vert ^{2}\right) \right].$$ Some fundamental inequalities for this quantum $f$-divergence in the case of convex functions are established. Applications for particular quantum divergence measures of interest are also provided.

Note

The author would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of the paper.

Article information

Source
Nihonkai Math. J., Volume 29, Number 1 (2018), 1-19.

Dates
Received: 13 October 2016
Revised: 15 February 2018
First available in Project Euclid: 6 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1549422080

Mathematical Reviews number (MathSciNet)
MR3908815

Zentralblatt MATH identifier
07063837

Subjects
Primary: 47A63: Operator inequalities 47A30: Norms (inequalities, more than one norm, etc.)
Secondary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 26D15: Inequalities for sums, series and integrals 26D10: Inequalities involving derivatives and differential and integral operators

Keywords
operator perspective convex functions operator inequalities arithmetic mean-geometric mean operator inequality relative operator entropy

Citation

Dragomir, Silvestru Sever. The Quadratic Quantum $f$-Divergence of Convex Functions and Matrices. Nihonkai Math. J. 29 (2018), no. 1, 1--19. https://projecteuclid.org/euclid.nihmj/1549422080


Export citation

References

  • T. Ando, Matrix Young inequalities, Oper. Theory Adv. Appl. 75 (1995), 33–38.
  • G. de Barra, Measure Theory and Integration, Ellis Horwood Ltd., 1981.
  • R. Bellman, Some inequalities for positive definite matrices, in: E.F. Beckenbach (Ed.), General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, Birkhäuser, Basel, 1980, pp. 89–90.
  • E. V. Belmega, M. Jungers and S. Lasaulce, A generalization of a trace inequality for positive definite matrices. Aust. J. Math. Anal. Appl. 7 (2010), Art. 26, 5 pp.
  • P. Cerone and S. S. Dragomir, Approximation of the integral mean divergence and $f$-divergence via mean results, Math. Comput. Modelling 42 (2005), 207–219.
  • P. Cerone, S. S. Dragomir and F. Österreicher, Bounds on extended $f$-divergences for a variety of classes. Kybernetika (Prague) 40 (2004), 745–756.
  • D. Chang, A matrix trace inequality for products of Hermitian matrices, J. Math. Anal. Appl. 237 (1999), 721–725.
  • L. Chen and C. Wong, Inequalities for singular values and traces, Linear Algebra Appl. 171 (1992), 109–120.
  • I. D. Coope, On matrix trace inequalities and related topics for products of Hermitian matrix, J. Math. Anal. Appl. 188 (1994), 999–1001.
  • I. Csiszár, Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten, (German) Magyar Tud. Akad. Mat. Kutató Int. Kö zl. 8 (1963), 85–108.
  • S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (2006), 471–478.
  • S. S. Dragomir, Some inequalities for $(m,M)$ -convex mappings and applications for the Csiszár $\Phi $ -divergence in information theory, Math. J. Ibaraki Univ. 33 (2001), 35–50.
  • S. S. Dragomir, Some inequalities for two Csiszá r divergences and applications, Mat. Bilten 25 (2001), 73–90.
  • S. S. Dragomir, An upper bound for the Csiszár f-divergence in terms of the variational distance and applications, Panamer. Math. J. 12 (2002), 43–54.
  • S. S. Dragomir, Upper and lower bounds for Csiszá r $f$-divergence in terms of Hellinger discrimination and applications, Nonlinear Anal. Forum 7 (2002), 1–13.
  • S. S. Dragomir, Bounds for $f$-divergences under likelihood ratio constraints, Appl. Math. 48 (2003), 205–223.
  • S. S. Dragomir, New inequalities for Csiszár divergence and applications, Acta Math. Vietnam. 28 (2003), 123–134.
  • S. S. Dragomir, A generalized $f$-divergence for probability vectors and applications, Panamer. Math. J. 13 (2003), 61–69.
  • S. S. Dragomir, Some inequalities for the Csiszár $\varphi $-divergence when $\varphi $ is an $L$ -Lipschitzian function and applications, Ital. J. Pure Appl. Math. 15 (2004), 57–76.
  • S. S. Dragomir, A converse inequality for the Csisz ár $\Phi $-divergence, Tamsui Oxf. J. Math. Sci. 20 (2004), 35–53.
  • S. S. Dragomir, Some general divergence measures for probability distributions, Acta Math. Hungar. 109 (2005), 331–345.
  • S. S. Dragomir, A refinement of Jensen's inequality with applications for $f$-divergence measures, Taiwanese J. Math. 14 (2010), 153–164.
  • S. S. Dragomir, A generalization of $f$ -divergence measure to convex functions defined on linear spaces, Commun. Math. Anal. 15 (2013), 1–14.
  • S. S. Dragomir, Inequalities for quantum $f$ -divergence of trace class operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 123. [Online http://rgmia.org/papers/v17/v17a123.pdf].
  • S. S. Dragomir, The quadratic weighted geometric mean for bounded linear operators in Hilbert spaces, Preprint, RGMIA Res. Rep. Coll. 19 (2016), Art. 145. [Online http://rgmia.org/papers/v19/v19a145.pdf].
  • S. Furuichi and M. Lin, Refinements of the trace inequality of Belmega, Lasaulce and Debbah, Aust. J. Math. Anal. Appl. 7 (2010), Art. 23, 4 pp.
  • F. Hiai, Fumio and D. Petz, From quasi-entropy to various quantum information quantities, Publ. Res. Inst. Math. Sci. 48 (2012), 525–542.
  • F. Hiai, M. Mosonyi, D. Petz and C. Bény, Quantum $f$-divergences and error correction, Rev. Math. Phys. 23 (2011), 691–747.
  • P. Kafka, F. Österreicher and I. Vincze, On powers of $f$-divergence defining a distance, Studia Sci. Math. Hungar. 26 (1991), 415–422.
  • H. D. Lee, On some matrix inequalities, Korean J. Math. 16 (2008), 565–571.
  • F. Liese and I. Vajda, Convex Statistical Distances, Teubuer – Texte zur Mathematik, Band 95, Leipzig, 1987.
  • L. Liu, A trace class operator inequality, J. Math. Anal. Appl. 328 (2007), 1484–1486.
  • S. Manjegani, Hölder and Young inequalities for the trace of operators, Positivity 11 (2007), 239–250.
  • H. Neudecker, A matrix trace inequality, J. Math. Anal. Appl. 166 (1992), 302–303.
  • F. Österreicher and I. Vajda, A new class of metric divergences on probability spaces and its applicability in statistics, Ann. Inst. Statist. Math. 55 (2003), 639–653.
  • D. Petz, From quasi-entropy, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 55 (2012), 81–92.
  • D. Petz, From $f$-divergence to quantum quasi-entropies and their use, Entropy 12 (2010), 304–325.
  • M. B. Ruskai, Inequalities for traces on von Neumann algebras, Commun. Math. Phys. 26 (1972), 280–289.
  • K. Shebrawi and H. Albadawi, Operator norm inequalities of Minkowski type, J. Inequal. Pure Appl. Math. 9 (2008), 1–10.
  • K. Shebrawi and H. Albadawi, Trace inequalities for matrices, Bull. Aust. Math. Soc. 87 (2013), 139–148.
  • B. Simon, Trace ideals and their applications, Cambridge University Press, Cambridge, 1979.
  • Z. Ulukök and R. Türkmen, On some matrix trace inequalities, J. Inequal. Appl. 2010, Art. ID 201486, 8 pp.
  • X. Yang, A matrix trace inequality, J. Math. Anal. Appl. 250 (2000), 372–374.
  • X. M. Yang, X. Q. Yang and K. L. Teo, A matrix trace inequality, J. Math. Anal. Appl. 263 (2001), 327–331.
  • Y. Yang, A matrix trace inequality, J. Math. Anal. Appl. 133 (1988), 573–574.