Banach Journal of Mathematical Analysis

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

Marcell Gaál

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1498097003

Digital Object Identifier
doi:10.1215/17358787-2017-0015

Mathematical Reviews number (MathSciNet)
MR3708527

Zentralblatt MATH identifier
06841252

Citation

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. https://projecteuclid.org/euclid.bjma/1498097003

References

• [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.