Abstract
For an arbitrary strictly convex function \(f\) defined on the non-negative real line we determine the structure of all transformations on the set of density operators which preserve the quantum \(f\)-divergence.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Notes
We mention that, as one of the referees pointed out, opposed to the conventions in mathematics, in physics the inner product is usually assumed to be conjugate-linear in its first variable and linear in the second. Accordingly, \(\langle A,B\rangle _\mathrm{HS }\) would be defined as \(\mathrm Tr A^*B\). However, in this paper we follow the common mathematical convention which is going to cause no confusion.
References
Ali, S.M., Silvey, S.D.: A general class of coefficients of divergence of one distribution from another. J. R. Stat. Soc. Ser. B 28, 131–142 (1966)
Csiszár, I.: Information type measure of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung. 2, 299–318 (1967)
Furuichi, S., Yanagi, K., Kuriyama, K.: Fundamental properties of Tsallis relative entropy. J. Math. Phys. 45, 4868–4877 (2004)
Hiai, F., Mosonyi, M., Petz, D., Bény, C.: Quantum \(f\)-divergences and error correction. Rev. Math. Phys. 23, 691–747 (2011)
Liese, F., Vajda, I.: Convex Statistical Distances. B.G. Teubner Verlagsgesellschaft, Leipzig (1987)
Molnár, L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Lecture Notes in Mathematics, vol. 1895. Springer, Berlin Heidelberg (2007)
Molnár, L.: Maps on states preserving the relative entropy. J. Math. Phys. 49, 032114 (2008)
Molnár, L., Szokol, P.: Maps on states preserving the relative entropy II. Linear Algebra Appl. 432, 3343–3350 (2010)
Niculescu, C., Persson, L.-E.:Convex Functions and Their Applications. A Contemporary Approach. CMC Books in Mathematics. Springer, New York (2006)
Petz, D.: Quasi-entropies for states of a von Neumann algebra. Publ. RIMS. Kyoto Univ. 21, 781–800 (1985)
Petz, D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23, 57–65 (1986)
Petz, D.: Quantum Information Theory and Quantum Statistics. Springer, Berlin (2008)
Petz, D.: From \(f\)-divergence to quantum quasi-entropies and their use. Entropy 12, 304–325 (2010)
Šemrl, P.: Comparability preserving maps on Hilbert space effect algebras. Comm. Math. Phys. 313, 375–384 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors were supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences. The first and second author were supported by the Hungarian Scientific Research Fund (OTKA) Reg.No. K81166 NK81402. The second and third author were supported by the TÁMOP-4.2.2/B-10/1-2010-0024 project co-financed by the European Union and the European Social Fund.
Rights and permissions
About this article
Cite this article
Molnár, L., Nagy, G. & Szokol, P. Maps on density operators preserving quantum \(f\)-divergences. Quantum Inf Process 12, 2309–2323 (2013). https://doi.org/10.1007/s11128-013-0528-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-013-0528-6