Skip to main content
SpringerLink
Log in

Quantifying the quantumness of ensembles via unitary similarity invariant norms

  • Research article
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

The quantification of the quantumness of a quantum ensemble has theoretical and practical significance in quantum information theory. We propose herein a class of measures of the quantumness of quantum ensembles using the unitary similarity invariant norms of the commutators of the constituent density operators of an ensemble. Rigorous proof shows that they share desirable properties for a measure of quantumness, such as positivity, unitary invariance, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Several specific examples illustrate the applications of these measures of quantumness in studying quantum information.

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. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge: Cambridge University Press, 2010

    Book  Google Scholar 

  2. L. Diósi, A Short Course in Quantum Information Theory: An Approach from Theoretical Physics, 2nd Ed., Berlin Heidelberg: Springer-Verlag, 2011

    Book  Google Scholar 

  3. S. Massar and S. Popescu, Optimal extraction of information from finite quantum ensembles, Phys. Rev. Lett. 74(8), 1259 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  4. C. A. Fuchs, Just two nonorthogonal quantum states, Quantum Communication, Computing, and Measurement 2, 11–16, Boston: Springer, 2002

    Google Scholar 

  5. C. A. Fuchs and M. Sasaki, Squeezing quantum information through a classical channel: Measuring the quantumness of a set of quantum states, Quantum Inf. Comput. 3, 377 (2003)

    MathSciNet  MATH  Google Scholar 

  6. C. A. Fuchs and M. Sasaki, The quantumness of a set of quantum states, arXiv: quant-ph/0302108

  7. M. Horodecki, P. Horodecki, R. Horodecki, and M. Piani, Quantumness of ensemble from no-broadcasting principle, Int. J. Quant. Inf. 04(01), 105 (2006)

    Article  Google Scholar 

  8. O. Oreshkov and J. Calsamiglia, Distinguishability measures between ensembles of quantum states, Phys. Rev. A 79(3), 032336 (2009)

    Article  ADS  Google Scholar 

  9. X. Zhu, S. Pang, S. Wu, and Q. Liu, The classicality and quantumness of a quantum ensemble, Phys. Lett. A 375(18), 1855 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  10. T. Ma, M. J. Zhao, Y. K. Wang, and S. M. Fei, Noncommutativity and local indistinguishability of quantum states, Sci. Rep. 4, 6336 (2014)

    Article  Google Scholar 

  11. S. Luo, N. Li, and X. Cao, Relative entropy between quantum ensembles, Period. Math. Hung. 59(2), 223 (2009)

    Article  MathSciNet  Google Scholar 

  12. S. Luo, N. Li, and W. Sun, How quantum is a quantum ensemble, Quantum Inform. Process. 9(6), 711 (2010)

    Article  MathSciNet  Google Scholar 

  13. S. Luo, N. Li, and S. Fu, Quantumness of quantum ensembles, Theor. Math. Phys. 169(3), 1724 (2011)

    Article  MathSciNet  Google Scholar 

  14. N. Li, S. Luo, and Y. Mao, Quantifying the quantumness of ensembles, Phys. Rev. A 96(2), 022132 (2017)

    Article  ADS  Google Scholar 

  15. B. Dakić, V. Vedral, and Č. Brukner, Necessary and sufficient condition for nonzero quantum discord, Phys. Rev. Lett. 105(19), 190502 (2010)

    Article  ADS  Google Scholar 

  16. S. Luo and S. Fu, Measurement-induced nonlocality, Phys. Rev. Lett. 106(12), 120401 (2011)

    Article  ADS  Google Scholar 

  17. M. L. Hu and H. Fan, Measurement-induced nonlocality based on trace norm, New J. hys. 17(3), 033004 (2015)

    ADS  Google Scholar 

  18. X. Zhan, Matrix Theory, Graduate Studies in Mathematics Vol. 147, American Mathematical Society, Providence, Rhode Island, 2013

    Google Scholar 

  19. Y. Peng, Y. Jiang, and H. Fan, Maximally coherent states and coherence-preserving operations, Phys. Rev. A 93(3), 032326 (2016)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  21. X. Yuan, H. Zhou, Z. Cao, and X. Ma, Intrinsic randomness as a measure of quantum coherence, Phys. Rev. A 92(2), 022124 (2015)

    Article  ADS  Google Scholar 

  22. X. F. Qi, T. Gao, and F. L. Yan, Measuring coherence with entanglement concurrence, J. Phys. A Math. Theor. 50(28), 285301 (2017)

    Article  MathSciNet  Google Scholar 

  23. S. Hill and W. K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett. 78(26), 5022 (1997)

    Article  ADS  Google Scholar 

  24. W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80(10), 2245 (1998)

    Article  ADS  Google Scholar 

  25. H. Ollivier and W. H. Zurek, Quantum discord: A measure of the quantumness of correlations, Phys. Rev. Lett. 88(1), 017901 (2001)

    Article  ADS  Google Scholar 

  26. L. Henderson and V. Vedral, Classical, quantum and total correlations, J. Phys. A Math. Gen. 34(35), 6899 (2001)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant Nos. 11371005 and 11475054 and the Hebei Natural Science Foundation under Grant Nos. A2016205145 and A2018205125.

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, China

    Xian-Fei Qi & Ting Gao

  2. College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang, 050024, China

    Feng-Li Yan

Authors
  1. Xian-Fei Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ting Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Feng-Li Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Ting Gao or Feng-Li Yan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qi, XF., Gao, T. & Yan, FL. Quantifying the quantumness of ensembles via unitary similarity invariant norms. Front. Phys. 13, 130309 (2018). https://doi.org/10.1007/s11467-018-0773-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-018-0773-3

Keywords

PACS numbers

Search

Navigation