Skip to main content
SpringerLink
Log in

Interplay between coherence and mixedness as well as its geometry for arbitrary two-qubit X-states

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Recently, a relationship between coherence and mixedness for an arbitrary d-dimensional quantum states has been built by Singh et al. (Phys Rev A 91:052115, 2015). However, whether this relationship holds under the action of decoherence remains to be further investigated. In this paper, we firstly study the geometry of coherence and mixedness for a class of two-qubit X-states and demonstrate new pictures and structures of trade-off between coherence and mixedness. At the same time, we also examine the dynamical behaviors of coherence, mixedness and their trade-off for both qubit–qubit system and environment–environment system where a two-qubit composite system is interacting with their own environmental channels. Different types of channels, such as amplitude-damping, phase-damping and bit-phase-flip channels with (non-)Markovian effects, are taken into consideration. Our results show that (i) Coherence can be completely transferred from the qubit–qubit system to the environment–environment system in the Markovian environment while in non-Markovian scenarios, the coherence between the qubit–qubit system and environment–environment system can be transferred each other. (ii) The decrease in the coherence is not always accompanied by an increase in the mixedness. (iii) The trade-off between coherence and mixedness still holds in the open system.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Matera, J.M., Egloff, D., Killoran, N., Plenio, M.B.: Coherent control of quantum systems as a resource theory. Quantum Sci. Technol. 1, 01LT01 (2016)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  6. Levi, F., Mintert, F.: A quantitative theory of coherent delocalization. New J. Phys. 16, 033007 (2014)

    Article  ADS  MATH  Google Scholar 

  7. Pires, D.P., Celeri, L.C., Soares-Pinto, D.O.: Geometric lower bound for a quantum coherence measure. Phys. Rev. A 91, 042330 (2015)

    Article  ADS  Google Scholar 

  8. Shao, L.H., Xi, Z., Fan, H., Li, Y.: The fidelity and trace norm distances for quantifying coherence. Phys. Rev. A 91, 042120 (2014)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. Du, S.P., Bai, Z.F.: The Wigner-Yanase information can increase under phase sensitive incoherent operations. Ann. Phys. 359, 136–140 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Girolami, D.: Observable measure of quantum coherence in finite dimensional systems. Phys. Rev. Lett. 113, 170401 (2014)

    Article  ADS  Google Scholar 

  12. Yao, Y., Xiao, X., Ge, L., Sun, C.P.: Quantum coherence in multipartite systems. Phys. Rev. A 92, 022112 (2015)

    Article  ADS  Google Scholar 

  13. Yu, X.D., Zhang, D.J., Liu, C.L., Tong, D.M.: Measure-independent freezing of quantum coherence. Phys. Rev. A 93, 060303 (2016)

    Article  ADS  Google Scholar 

  14. Napoli, C., Bromley, T., Cianciaruso, R.M., Piani, M., Johnston, N., Adesso, G.: Robustness of coherence: an operational and observable measure of quantum coherence. Phys. Rev. Lett. 116, 150502 (2016)

    Article  ADS  Google Scholar 

  15. Liu, X.B., Tian, Z.H., Wang, J.C., Jing, J.L.: Protecting quantum coherence of two-level atoms from vacuum fluctuations of electromagnetic field. Ann. Phys. 366, 102 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Wang, J.C., Tian, Z.H., Jing, J.L., Fan, H.: Irreversible degradation of quantum coherence under relativistic motion. Phys. Rev. A 93, 062105 (2016)

    Article  ADS  Google Scholar 

  17. Guo, Y.N., Tian, Q.L., Zeng, K., Li, Z.D.: Quantum coherence of two-qubit over quantum channels with memory. Quantum Inf. Process. 16, 310 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Lin, D.P., Zou, H.M., Yang, J.H.: Based-nonequilibrium-environment non-Markovianity, quantum Fisher information and quantum coherence. Phys. Scr. 95, 015103 (2020)

    Article  ADS  Google Scholar 

  19. Huang, Z.M., Situ, H.Z.: Non-Markovian dynamics of quantum coherence of two-level system driven by classical field. Quantum Inf. Process. 16, 222 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Situ, H.Z., Hu, X.Y.: Dynamics of relative entropy of coherence under Markovian channels. Quantum Inf. Process. 15, 4649 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Aaronson, B., Lo Franco, R., Adesso, G.: Comparative investigation of the freezing phenomena for quantum correlations under nondissipative decoherence. Phys. Rev. A 88, 012120 (2013)

    Article  ADS  Google Scholar 

  22. Cianciaruso, M., Bromley, T.R., Roga, W., Lo Franco, R., Adesso, G.: Universal freezing of quantum correlations within the geometric approach. Sci. Rep. 5, 10177 (2015)

    Article  ADS  Google Scholar 

  23. Bromley, T.R., Cianciaruso, M., Adesso, G.: Frozen quantum coherence. Phys. Rev. Lett. 114, 210401 (2015)

    Article  ADS  Google Scholar 

  24. Singh, U., Nath Bera, M., Dhar, H.S., Pati, A.K.: Maximally coherent mixed states: complementarity between maximal coherence and mixedness. Phys. Rev. A 91, 052115 (2015)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Horodecki, R., Horodecki, M.: Information-theoretic aspects of inseparability of mixed states. Phys. Rev. A 54, 1838 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Lang, M.D., Caves, C.M.: Quantum Discord and the Geometry of Bell-Diagonal States. Phys. Rev. Lett. 105, 150501 (2010)

    Article  ADS  Google Scholar 

  28. Li, B., Wang, Z.X., Fei, S.M.: Quantum discord and geometry for a class of two-qubit states. Phys. Rev. A 83, 022321 (2011)

    Article  ADS  Google Scholar 

  29. Yao, Y., Li, H.W., Yin, Z.Q., Han, Z.F.: Geometric interpretation of the geometric discord. Phys. Lett. A 376, 358 (2012)

    Article  ADS  MATH  Google Scholar 

  30. Huang, Z., Qiu, D., Mateus, P.: Geometry and dynamics of one-norm geometric quantum discord. Quantum Inf. Process. 15, 301 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Wang, Y.K., Ma, T., Li, B., Wang, Z.X.: One-way information deficit and geometry for a class of two-qubit states. Commun. Theor. Phys. 59, 540 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Wang, Y.K., Shao, L.H., Ge, L.Z., Fei, S.M., Wang, Z.X.: Geometry of quantum coherence for two qubit X states. Int. J. Theor. Phys. 58, 2372 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Peters, N.A., Wei, T.C., Kwiat, P.G.: Mixed-state sensitivity of several quantum-information benchmarks. Phys. Rev. A 70, 052309 (2004)

    Article  ADS  Google Scholar 

  34. Maziero, J., Werlang, T., Fanchini, F.F., Celeri, L.C., Serra, R.M.: System-reservoir dynamics of quantum and classical correlations. Phys. Rev. A 81, 022116 (2010)

    Article  ADS  Google Scholar 

  35. Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  36. Bellomo, B., Lo Franco, R., Compagno, G.: Non-Markovian effects on the dynamics of entanglement. Phys. Rev. Lett. 99, 160502 (2007)

    Article  ADS  Google Scholar 

  37. Yu, T., Eberly, J.: Entanglement evolution in a non-Markovian environment. Opt. Commun. 283, 676 (2010)

    Article  ADS  Google Scholar 

  38. Paulson, K.G., Panwar Ekta Banerjee, S., Srikanth, R.: Hierarchy of quantum correlations under non-Markovian dynamics. Quantum Inf. Process. 20, 141 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  39. Milz, S., Kim, M., Pollock, F.A., Modi, K.: Completely positive divisibility does not mean Markovianity. Phys. Rev. Lett. 123, 040401 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  40. Utagi, S., Srikanth, R., Banerjee, S.: Temporal self-similarity of quantum dynamical maps as a concept of memorylessness. Sci. Rep. 10, 15049 (2020)

    Article  Google Scholar 

  41. Daffer, S., W\(\acute{o}\)dkiewicz, K., Cresser, J.D., McIver, J.K.: Depolarizing channel as a completely positive map with memory. Phys. Rev. A 70, 010304 (2004)

  42. Dixit, K., Naikoo, J., Banerjee, S., Alok, A.K.: Study of coherence and mixedness in meson and neutrino systems. Eur. Phys. J. C 79, 96 (2019)

    Article  ADS  Google Scholar 

  43. Bhattacharya, S., Banerjee, S., Pati, A.K.: Evolution of coherence and non-classicality under global environmental interaction. Quantum Inf. Proc. 17, 236 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Kurashvili, P., Chotorlishvili, L., Kouzakov, K.A., Studenikin, A.I.: Coherence and mixedness of neutrino oscillations in a magnetic field. Eur. Phys. J. C 81, 323 (2021)

    Article  ADS  Google Scholar 

  45. Naikoo, J., Dutta, S., Banerjee, S.: Facets of quantum information under non-Markovian evolution. Phys. Rev. A 99, 042128 (2019)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work is supported by the Scientific Research Project of Hunan Province Department of Education (Grant No.19B060) and the Natural Science Foundation of Hunan Province (Grant No.2021JJ30757). Y N Guo is supported by Training Program for Excellent Young Innovators of Changsha (kq2009076 and kq2106029).

Author information

Authors and Affiliations

  1. Department of Electronic and Communication Engineering, Changsha University, Changsha, 410022, Hunan, People’s Republic of China

    You-neng Guo, Xin Wang & Xiang-jun Chen

Authors
  1. You-neng Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiang-jun Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to You-neng Guo.

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

Guo, Yn., Wang, X. & Chen, Xj. Interplay between coherence and mixedness as well as its geometry for arbitrary two-qubit X-states. Quantum Inf Process 21, 149 (2022). https://doi.org/10.1007/s11128-022-03495-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-022-03495-9

Keywords

Search

Navigation