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Maps on positive operators preserving Rényi type relative entropies and maximal f-divergences

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Abstract

In this paper, we deal with two quantum relative entropy preserver problems on the cones of positive (either positive definite or positive semidefinite) operators. The first one is related to a quantum Rényi relative entropy like quantity which plays an important role in classical–quantum channel decoding. The second one is connected to the so-called maximal f-divergences introduced by D. Petz and M. B. Ruskai who considered this quantity as a generalization of the usual Belavkin–Staszewski relative entropy. We emphasize in advance that all the results are obtained for finite-dimensional Hilbert spaces.

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Notes

  1. For the sake of simplicity in the Introduction we take the freedom to define the corresponding divergences only on the set of positive definite operators. The extensions for general (not necessarily invertible) positive semidefinite operators are discussed in Sect. 2.

  2. On the set of positive definite operators the case where f is operator monotone decreasing has been already solved before by Molnár [14].

  3. Here, trace preserving property means that \({\text {Tr}}\Gamma (p)= \sum _{i=1}^{d} p_i\) holds for any probability distribution p.

References

  1. Bhatia, R.: Matrix Analysis. Springer, New York (1997)

    Book  MATH  Google Scholar 

  2. Csiszár, I.: Information type measure of difference of probability distributions and indirect observations. Stud. Sci. Math. Hung. 2, 299–318 (1967)

    MathSciNet  MATH  Google Scholar 

  3. Dolinar, G., Molnár, L.: Automorphisms for the logarithmic product of positive semidefinite operators. Linear Multilinear Algebra 61(2), 161–169 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gaál, M.: Maps preserving a new version of quantum \(f\)-divergence. Banach J. Math. Anal. 11(4), 744–763 (2017)

    Article  MathSciNet  Google Scholar 

  5. Gaál, M., Molnár, L.: Transformations on density operators and on positive definite operators preserving the quantum Rényi divergence. Period. Math. Hung. 74, 88–107 (2017)

    Article  MATH  Google Scholar 

  6. Hansen, F., Pedersen, G.K.: Jensen’s inequality for operators and Löwner’s theorem. Math. Ann. 258, 229–241 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hiai, F., Mosonyi, M.: Different quantum \(f\)-divergences and the reversibility of quantum operations. Rev. Math. Phys. (2017). https://doi.org/10.1142/S0129055X17500234

    MathSciNet  MATH  Google Scholar 

  8. Hiai, F., Mosonyi, M., Petz, D., Bény, C.: Quantum \(f\)-divergences and error correction. Rev. Math. Phys. 23, 691–747 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kraus, F.: Über konvexe Matrixfunktionen. Math. Z. 41, 18–42 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. Matsumoto, K.: A new quantum version of \(f\)-divergence. Preprint, arXiv:1311.4722v2

  12. Molnár, L.: Maps on states preserving the relative entropy. J. Math. Phys. 49, 032114 (2008)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  13. Molnár, L.: Order automorphisms on positive definite operators and a few applications. Linear Algebra Appl. 434, 2158–2169 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Molnár, L.: Two characterizations of unitary–antiunitary similarity transformations of positive definite operators on a finite dimensional Hilbert space. Ann. Univ. Sci. 58, 83–93 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Molnár, L., Nagy, G., Szokol, P.: Maps on density operators preserving quantum \(f\)-divergences. Quantum Inf. Process. 12, 2309–2323 (2013)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  16. Molnár, L., Pitrik, J., Virosztek, D.: Maps on positive definite matrices preserving Bregmann and Jensen divergences. Linear Algebra Appl. 495, 174–189 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Molnár, L., Szokol, P.: Transformations preserving norms of means of positive operators and nonnegative functions. Integral Equ. Oper. Theory 83, 271–290 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mosonyi, M., Ogawa, T.: Strong converse exponent for classical–quantum channel coding. Commun. Math. Phys. 335, 373–426 (2017)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  19. Müller-Lennert, M., Dupuis, F., Szehr, O., Fehr, S., Tomamichel, M.: On quantum Rényi entropies: a new generalization and some properties. J. Math. Phys. 54, 122203 (2013)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  20. Petz, D.: Quasi-entropies for states of a von Neumann algebra. Publ. RIMS. Kyoto Univ. 21, 787–800 (1985)

    Article  MATH  Google Scholar 

  21. Petz, D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23, 57–65 (1986)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  22. Petz, D., Ruskai, M.B.: Contraction of generalized relative entropy under stochastic mappings on matrices. Infin. Dimens. Anal. Quantum Probab. 1, 83–89 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Academic Press, Boston (1990)

    MATH  Google Scholar 

  24. Virosztek, D.: Maps on quantum states preserving Bregman and Jensen divergences. Lett. Math. Phys. 106, 1217–1234 (2016)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  25. Virosztek, D.: Quantum \(f\)-divergence preserving maps on positive semidefinite operators acting on finite dimensional Hilbert spaces. Linear Algebra Appl. 501, 242–253 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the anonymous referee for carefully reading the manuscript and for a comment which simplified the last part of the proof of Theorem 1. The authors were supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office NKFIH Reg. No. K115383.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, 6720, Hungary

    Marcell Gaál

  2. MTA-DE “Lendület” Functional Analysis Research Group, Institute of Mathematics, University of Debrecen, PO. Box 400, Debrecen, 4002, Hungary

    Marcell Gaál & Gergő Nagy

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  1. Marcell Gaál
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  2. Gergő Nagy
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Correspondence to Gergő Nagy.

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Gaál, M., Nagy, G. Maps on positive operators preserving Rényi type relative entropies and maximal f-divergences. Lett Math Phys 108, 425–443 (2018). https://doi.org/10.1007/s11005-017-1021-4

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