Skip to main content
SpringerLink
Log in

Monotonicity of skew information and its applications in quantum resource theory

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

An alternative proof about the monotonicity of skew information via operator algebra approach is formulated. Furthermore, the strong monotonicity of skew information under particular quantum Trace-Preserving and Completely Positive maps is confirmed. A family of new quantum resource measures is introduced if the resource can be characterized by a resource destroying map and the free operation should be also modified. The proposed measure is easy-calculating and applicable to the quantum coherence resource theory as well as the quantum asymmetry theory. The operational interpretation needs to be further investigated.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Wigner, E.P., Yanase, M.M.: Information contents of distributions. Proc. Natl. Acad. Sci. 49(6), 910–918 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  2. Wigner, E.P., Yanase, M.M.: On the positive semidefinite nature of a certain matrix expression. Can. J. Math 16, 397–406 (1964)

    Article  MathSciNet  Google Scholar 

  3. Luo, S.: Wigner–Yanase skew information and uncertainty relations. Phys. Rev. Lett. 91(18), 180403 (2003)

    Article  ADS  Google Scholar 

  4. Luo, S., Zhang, Q.: Skew information decreases under quantum measurements. Theor. Math. Phys. 151(1), 529–538 (2007)

    Article  Google Scholar 

  5. Luo, S., Sun, Y.: Coherence and complementarity in state-channel interaction. Phys. Rev. A 98(1), 012113 (2018)

    Article  ADS  Google Scholar 

  6. Chitambar, E., Gour, G.: Quantum resource theories (2018). arXiv:1806.06107

  7. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  8. Winter, A., Yang, D.: Operational resource theory of coherence. Phys. Rev. Lett. 116(12), 120404 (2016)

    Article  ADS  Google Scholar 

  9. Gour, G., Spekkens, R.W.: The resource theory of quantum reference frames: manipulations and monotones. New J. Phys. 10(3), 033023 (2008)

    Article  ADS  Google Scholar 

  10. Brandão, F.G.S.L., Gour, G.: Reversible framework for quantum resource theories. Phys. Rev. Lett. 115(7), 070503 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  11. Coecke, B., Fritz, T., Spekkens, R.W.: A mathematical theory of resources. Inf. Comput. 250, 59–86 (2016)

    Article  MathSciNet  Google Scholar 

  12. Chitambar, E., Gour, G.: Comparison of incoherent operations and measures of coherence. Phys. Rev. A 94(5), 052336 (2016)

    Article  ADS  Google Scholar 

  13. Liu, Z.-W., Hu, X., Lloyd, S.: Resource destroying maps. Phys. Rev. Lett. 118(6), 060502 (2017)

    Article  ADS  Google Scholar 

  14. Størmer, E.: Positive Linear Maps of Operator Algebras. Springer, Berlin (2012)

    MATH  Google Scholar 

  15. Hiai, F., Petz, D.: From quasi-entropy to various quantum information quantities. Publ. Res. Inst. Math. Sci. 48(3), 525–542 (2012)

    Article  MathSciNet  Google Scholar 

  16. Bhatia, R.: Matrix Analysis, vol. 169. Springer, Berlin (2013)

    MATH  Google Scholar 

  17. Jençová, A., Ruskai, M.B.: A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality. Rev. Math. Phys. 22(09), 1099–1121 (2010)

    Article  MathSciNet  Google Scholar 

  18. Gour, G.: Quantum resource theories in the single-shot regime. Phys. Rev. A 95(6), 062314 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  19. Marvian Mashhad, I.: Symmetry, asymmetry and quantum information (2012)

  20. Marvian, I., Spekkens, R.W.: The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations. New J. Phys. 15(3), 033001 (2013)

    Article  ADS  Google Scholar 

  21. Streltsov, A., Adesso, G., Plenio, M.B.: Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89(4), 041003 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  22. Marvian, I., Spekkens, R.W.: How to quantify coherence: distinguishing speakable and unspeakable notions. Phys. Rev. A 94(5), 052324 (2016)

    Article  ADS  Google Scholar 

  23. Chang-shui, Y.: Quantum coherence via skew information and its polygamy. Phys. Rev. A 95(4), 042337 (2017)

    Article  ADS  Google Scholar 

  24. Vershynina, A.: On quantum quasi-relative entropy (2018). arXiv:1810.00391

  25. Zhao, H., Chang-shui, Y.: Coherence measure in terms of the tsallis relative \(\alpha \) entropy. Sci. Rep. 8(1), 299 (2018)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The author is very grateful to professor Shunlong Luo, professor Shaoming Fei and Wei Xie for insightful discussions.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Weijing Li

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Weijing Li

  3. UTS-AMSS Joint Research Laboratory for Quantum Computation and Quantum Information Processing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Weijing Li

Authors
  1. Weijing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weijing Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, W. Monotonicity of skew information and its applications in quantum resource theory. Quantum Inf Process 18, 166 (2019). https://doi.org/10.1007/s11128-019-2284-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-019-2284-8

Keywords

Search

Navigation