Banach Journal of Mathematical Analysis

Maps preserving a new version of quantum $f$-divergence

Marcell Gaál

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


For an arbitrary nonaffine operator convex function defined on the nonnegative real line and satisfying $f(0)=0$, we characterize the bijective maps on the set of all positive definite operators preserving a new version of quantum $f$-divergence. We also determine the structure of all transformations leaving this quantity invariant on quantum states for any strictly convex functions with the properties $f(0)=0$ and $\lim_{x\to\infty}f(x)/x=\infty$. Finally, we derive the corresponding result concerning those transformations on the set of positive semidefinite operators. We emphasize that all the results are obtained for finite-dimensional Hilbert spaces.

Article information

Banach J. Math. Anal. Volume 11, Number 4 (2017), 744-763.

Received: 6 July 2016
Accepted: 23 October 2016
First available in Project Euclid: 22 June 2017

Permanent link to this document

Digital Object Identifier

Primary: 47B49: Transformers, preservers (operators on spaces of operators)
Secondary: 47N50: Applications in the physical sciences

preservers positive definite operators density operators quantum states relative entropy


Gaál, Marcell. Maps preserving a new version of quantum f -divergence. Banach J. Math. Anal. 11 (2017), no. 4, 744--763. doi:10.1215/17358787-2017-0015.

Export citation


  • [1] P. Busch and S. P. Gudder, Effects as functions on projective Hilbert space, Lett. Math. Phys. 47 (1999), no. 4, 329–337.
  • [2] E. Carlen, “Trace inequalities and quantum entropy: An introductory course” in Entropy and the Quantum (Tucson, 2009), Contemp. Math. 529, Amer. Math. Soc., Providence, 2010, 73–140.
  • [3] G. Chevalier, “Wigner’s theorem and its generalizations” in Handbook of Quantum Logic and Quantum Structures, Elsevier, Amsterdam, 2007, 429–475.
  • [4] I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Studia. Sci. Math. Hungar. 2 (1967), 299–318.
  • [5] S. S. Dragomir, A new quantum $f$-divergence for trace class operators in Hilbert spaces, Entropy 16 (2014), no. 11, 5853–5875.
  • [6] M. Gaál and L. Molnár, Transformations on density operators and on positive definite operators preserving the quantum Rényi divergence, Period. Math. Hungar. 74 (2017), no. 1, 88–107.
  • [7] Gy. P. Gehér, An elementary proof for the non-bijective version of Wigner’s theorem, Phys. Lett. A 378 (2014), no. 30–31, 2054–2057.
  • [8] M. Győry, A new proof of Wigner’s theorem, Rep. Math. Phys. 54 (2004), no. 2, 159–167.
  • [9] F. Hansen and G. K. Pedersen, Jensen’s inequality for operators and Löwner’s theorem, Math. Ann. 258 (1982), no. 3, 229–241.
  • [10] F. Hiai, M. Mosonyi, D. Petz, and C. Bény, Quantum $f$-divergences and error correction, Rev. Math. Phys. 23 (2011), no. 7, 691–747.
  • [11] F. Kraus, Über konvexe matrixfuntionen, Math. Z. 41 (1936), no. 1, 18–42.
  • [12] K. Matsumoto, A new quantum version of f-divergence, preprint, arXiv:1311.4722v3 [quant-ph].
  • [13] L. Molnár, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lecture Notes in Math. 1895, Berlin, Springer, 2007.
  • [14] L. Molnár, Maps on states preserving the relative entropy, J. Math. Phys. 49 (2008), no. 3, art. ID 032114.
  • [15] L. Molnár, Order automorphisms on positive definite operators and a few applications, Linear Algebra Appl. 434 (2011), no. 10, 2158–2169.
  • [16] L. Molnár, Two characterizations of unitary-antiunitary similarity transformations of positive definite operators on a finite-dimensional Hilbert space, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 58 (2015), 83–93.
  • [17] L. Molnár and G. Nagy, Isometries and relative entropy preserving maps on density operators, Linear Multilinear Algebra 60 (2012), no. 1, 93–108.
  • [18] L. Molnár, G. Nagy, and P. Szokol, Maps on density operators preserving quantum f-divergences, Quantum Inf. Process. 12 (2013), no. 7, 2309–2323.
  • [19] L. Molnár and P. Szokol, Maps on states preserving the relative entropy, II, Linear Algebra Appl. 432 (2010), no. 12, 3343–3350.
  • [20] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum Rényi entropies: A new generalization and some properties, J. Math. Phys. 54 (2013), no. 12, art. ID 122203.
  • [21] M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, Berlin, 1993.
  • [22] D. Petz, Quasientropies for states of a von Neumann algebra, Publ. Res. Inst. Math. Sci. 21(1985), no. 4, 787–800.
  • [23] D. Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys. 23 (1986), no. 1, 57–65.
  • [24] D. Virosztek, Maps on quantum states preserving Bregman and Jensen divergences, Lett. Math. Phys. 106 (2016), no. 9, 1217–1234.
  • [25] D. Virosztek, Quantum f-divergence preserving maps on positive semidefinite operators acting on finite dimensional Hilbert spaces, Linear Algebra Appl. 501 (2016), 242–253.