Skip to main content
SpringerLink
Log in

Measure of genuine coherence based of quasi-relative entropy

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We present a genuine coherence measure based on a quasi-relative entropy as a difference between quasi-entropies of the dephased and the original states. The measure satisfies non-negativity and monotonicity under genuine incoherent operations (GIO). It is strongly monotone under GIO in two and three dimensions, or for pure states in any dimension, making it a genuine coherence monotone. We provide a bound on the error term in the monotonicity relation under GIO in terms of the trace distance between the original and the dephased states. Moreover, the lower bound on the coherence measure can also be calculated in terms of this trace distance.

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. Aberg, J.: Catalytic coherence. Phys. Rev. Lett. 113, 150402 (2014)

    Article  ADS  Google Scholar 

  2. Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113(14), 140401 (2014)

    Article  ADS  Google Scholar 

  3. Carlen, E.A., Vershynina, A.: Recovery and the data processing inequality for quasi-entropies. IEEE Trans. Inf. Theory 64(10), 6929–6938 (2018)

    Article  MathSciNet  Google Scholar 

  4. Chitambar, E., Gour, G.: Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence. Phys. Rev. Lett. 117(3), 030401 (2016)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  6. Chitambar, E., Hsieh, M.H.: Relating the resource theories of entanglement and quantum coherence. Phys. Rev. Lett. 117(2), 020402 (2016)

    Article  ADS  Google Scholar 

  7. Csiszár, I.: Information-type measures of difference of probability distributions and indirect observation. Studia Scientiarum Mathematicarum Hungarica 2, 229–318 (1967)

    MathSciNet  Google Scholar 

  8. Cwikliski, P., Studziski, M., Horodecki, M., Oppenheim, J.: Limitations on the evolution of quantum coherences: towards fully quantum second laws of thermodynamics. Phys. Rev. Lett. 115(21), 210403 (2015)

    Article  ADS  Google Scholar 

  9. De Vicente, J.I., Streltsov, A.: Genuine quantum coherence. J. Phys. A: Math. Theor. 50(4), 045301 (2016)

    Article  MathSciNet  Google Scholar 

  10. Du, S., Bai, Z., Guo, Y.: Conditions for coherence transformations under incoherent operations. Phys. Rev. A 91(5), 052120 (2015)

    Article  ADS  Google Scholar 

  11. Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131(6), 2766 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  12. Hardy, G.H., Littlewood, J.E., Pólya, G.: Some simple inequalities satisfied by convex function. Messenger Math. 58, 145–152 (1929)

    MATH  Google Scholar 

  13. Hiai, F., Mosonyi, M., Petz, D., Bény, C.: Quantum \(f\)-divergences and error correction. Rev. Math. Phys. 23(07), 691–747 (2011)

    Article  MathSciNet  Google Scholar 

  14. Hiai, F., Mosonyi, M.: Different quantum \(f\)-divergences and the reversibility of quantum operations. Rev. Math. Phys. 29(07), 1750023 (2017)

    Article  MathSciNet  Google Scholar 

  15. Hu, M.L., Hu, X., Wang, J., Peng, Y., Zhang, Y.R., Fan, H.: Quantum coherence and geometric quantum discord. Phys. Rep. 762, 1–100 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  16. Lesniewski, A., Ruskai, M.B.: Monotone Riemannian metrics and relative entropy on noncommutative probability spaces. J. Math. Phys. 40(11), 5702–5724 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  17. Lostaglio, M., Jennings, D., Rudolph, T.: Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nat. Commun. 6, 6383 (2015)

    Article  ADS  Google Scholar 

  18. Ma, J., Yadin, B., Girolami, D., Vedral, V., Gu, M.: Converting coherence to quantum correlations. Phys. Rev. Lett. 116(16), 160407 (2016)

    Article  ADS  Google Scholar 

  19. Marshall, A.W., Olkin, I., Arnold, B.C.: Doubly stochastic matrices. In Inequalities: Theory of Majorization and Its Applications. pp. 29-77, Springer, New York, NY (2010)

  20. Piani, M., Cianciaruso, M., Bromley, T.R., Napoli, C., Johnston, N., Adesso, G.: Robustness of asymmetry and coherence of quantum states. Phys. Rev. A 93(4), 042107 (2016)

    Article  ADS  Google Scholar 

  21. Petz, D.: Quasi-entropies for states of a von Neumann algebra. Publ. Res. Inst. Math. Sci. 21(4), 787–800 (1985)

    Article  MathSciNet  Google Scholar 

  22. Petz, D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys. 23(1), 57–65 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  23. Radhakrishnan, C., Parthasarathy, M., Jambulingam, S., Byrnes, T.: Distribution of quantum coherence in multipartite systems. Phys. Rev. Lett. 116(15), 150504 (2016)

    Article  ADS  Google Scholar 

  24. Rana, S., Parashar, P., Lewenstein, M.: Trace-distance measure of coherence. Phys. Rev. A 93(1), 012110 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  25. Rastegin, A.E.: Quantum-coherence quantifiers based on the Tsallis relative \(\alpha \) entropies. Phys. Rev. A 93(3), 032136 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  26. Rebentrost, P., Mohseni, M., Aspuru-Guzik, A.: Role of quantum coherence and environmental fluctuations in chromophoric energy transport. J. Phys. Chem. B 113, 9942 (2009)

    Article  Google Scholar 

  27. Schur, I.: Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungsberichte der Berliner Mathematischen Gesellschaft 22(9–20), 51 (1923)

    MATH  Google Scholar 

  28. Scully, M.O., Zubairy, M.S.: Quantum Optics (Cambridge.) Ch, 4, 17 (1997)

  29. Shao, L.H., Xi, Z., Fan, H., Li, Y.: Fidelity and trace-norm distances for quantifying coherence. Phys. Rev. A 91(4), 042120 (2015)

    Article  ADS  Google Scholar 

  30. 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 

  31. Streltsov, A., Singh, U., Dhar, H.S., Bera, M.N., Adesso, G.: Measuring quantum coherence with entanglement. Phys. Rev. Lett. 115(2), 020403 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  32. Tan, K.C., Kwon, H., Park, C.Y., Jeong, H.: Unified view of quantum correlations and quantum coherence. Phys. Rev. A 94(2), 022329 (2016)

    Article  ADS  Google Scholar 

  33. Tomamichel, M., Colbeck, R., Renner, R.: A fully quantum asymptotic equipartition property. IEEE Trans. Inf. Theory 55(12), 5840–5847 (2009)

    Article  MathSciNet  Google Scholar 

  34. Virosztek, D.: Quantum entropies, relative entropies, and related preserver problems (2016)

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

    Article  ADS  Google Scholar 

  36. Witt, B., Mintert, F.: Stationary quantum coherence and transport in disordered networks. New J. Phys. 15, 093020 (2013)

    Article  ADS  Google Scholar 

  37. Yadin, B., Ma, J., Girolami, D., Gu, M., Vedral, V.: Quantum processes which do not use coherence. Phys. Rev. X 6(4), 041028 (2016)

    Google Scholar 

  38. Yu, X.D., Zhang, D.J., Xu, G.F., Tong, D.M.: Alternative framework for quantifying coherence. Phys. Rev. A 94(6), 060302 (2016)

    Article  ADS  Google Scholar 

  39. Zhao, H., Yu, C.S.: Coherence measure in terms of the Tsallis relative \(\alpha \) entropy. Sci. Rep. 8(1), 299 (2018)

    Article  ADS  Google Scholar 

  40. Zhu, H., Ma, Z., Cao, Z., Fei, S.M., Vedral, V.: Operational one-to-one mapping between coherence and entanglement measures. Phys. Rev. A 96(3), 032316 (2017)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

A. V. is supported by NSF grants DMS-1812734 and DMS-2105583.

Author information

Authors and Affiliations

  1. Department of Mathematics, Philip Guthrie Hoffman Hall, University of Houston, 3551 Cullen Blvd., Houston, TX, 77204-3008, USA

    Anna Vershynina

Authors
  1. Anna Vershynina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Vershynina.

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

Vershynina, A. Measure of genuine coherence based of quasi-relative entropy. Quantum Inf Process 21, 184 (2022). https://doi.org/10.1007/s11128-022-03531-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-022-03531-8

Search

Navigation