Skip to main content
Log in

Transformations on density operators and on positive definite operators preserving the quantum Rényi divergence

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

In a certain sense we generalize the recently introduced and extensively studied notion called quantum Rényi divergence (also called ”sandwiched Rényi relative entropy”) and describe the structures of corresponding symmetries. More precisely, we characterize all transformations on the set of density operators which leave our new general quantity invariant and also determine the structure of all bijective transformations on the cone of positive definite operators which preserve the quantum Rényi divergence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. E. Carlen, Trace inequalities and quantum entropy: an introductory course, in Entropy and the Quantum. Contemporary Mathematics, vol. 529 (American Mathematical Society, Providence, 2010), pp. 73–140

  2. N. Datta, F. Leditzky, A limit of the quantum Rényi divergence. J. Phys. A 47, 045304 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Dupuis, Chain rules for quantum Rényi entropies. J. Math. Phys. 56, 022203 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. R.L. Frank, E.H. Lieb, Monotonicity of a relative Rényi entropy. J. Math. Phys. 54, 122201 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities (Cambridge University Press, 1934)

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

    Article  MathSciNet  MATH  Google Scholar 

  7. S.M. Lin, M. Tomamichel, Investigating properties of a family of quantum Rényi divergences. Quantum Inf. Process. 14, 1501–1512 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Molnár, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Lecture Notes in Mathematics, vol. 1895 (Springer, Berlin Heidelberg, 2007)

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

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Molnár, G. Nagy, Isometries and relative entropy preserving maps on density operators. Linear and Multilinear Algebra 60, 93–108 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Molnár, P. Szokol, Maps on states preserving the relative entropy II. Linear Algebra Appl. 432, 3343–3350 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Mosonyi, T. Ogawa, Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies. Commun. Math. Phys. 334, 1617–1648 (2015)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. M.M. Wilde, A. Winter, D. Yang, Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy. Commun. Math. Phys. 331, 593–622 (2014)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the referee of the paper for the particularly careful reading of the manuscript and for making several useful comments. 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 (Grant No. K115383).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lajos Molnár.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1007/s10998-016-0174-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-016-0174-8

Keywords

Navigation