Skip to main content
SpringerLink
Log in

Quantum coherence in mutually unbiased bases

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We investigate the \(l_{1}\) norm of coherence of quantum states in mutually unbiased bases. We find that the sum of squared \(l_{1}\) norm of coherence of single qubit mixed state is less than two. We derive that the \(l_{1}\) norms of coherence of three classes of X states in nontrivial mutually unbiased bases for 4-dimensional Hilbert space are equal. We propose the concept of “autotensor of mutually unbiased basis (AMUB)” by the tensor of mutually unbiased bases and depict the level surface of the constant sum of the \(l_{1}\) norm of coherence of Bell-diagonal states in AMUB. We find that the \(l_{1}\) norms of coherence of Werner states and isotropic states in AMUB are equal, respectively.

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

Similar content being viewed by others

References

  1. Jha, P.K., Mrejen, M., Kim, J., Wu, C., Wang, Y., Rostovtsev, Y.V., Zhang, X.: Coherence-driven topological transition in quantum metamaterials. Phys. Rev. Lett. 116(16), 165502 (2016)

    Article  ADS  Google Scholar 

  2. Bagan, E., Bergou, J.A., Cottrell, S.S., Hillery, M.: Relations between coherence and path information. Phys. Rev. Lett. 116(16), 160406 (2016)

    Article  ADS  Google Scholar 

  3. Kammerlander, P., Anders, J.: Coherence and measurement in quantum thermodynamics. Sci. Rep. 6(1), 22174 (2016)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Sudarshan, E.C.G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10(7), 277 (1963)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Frank, B., Amy, N.: Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  7. Giovannetti, V., Lloyd, S., Maccone, L.: Quantum-enhanced measurements: beating the standard quantum limit. Science 306(5700), 1330 (2004)

    Article  ADS  Google Scholar 

  8. Demkowicz-Dobrzański, R., Maccone, L.: Using entanglement against noise in quantum metrology. Phys. Rev. Lett. 113(25), 250801 (2014)

    Article  ADS  Google Scholar 

  9. Giovannetti, V., Lloyd, S., Maccone, L.: Advances in quantum metrology. Nat. Photonics 5(4), 222 (2011)

    Article  ADS  Google Scholar 

  10. Lostaglio, M., Jennings, D., Rudolph, T.: Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nat. Commun. 6(5278), 6383 (2011)

    ADS  Google Scholar 

  11. Lostaglio, M., Korzekwa, K., Jennings, D., Rudolph, T.: Quantum coherence, time-translation symmetry and thermodynamics. Phys. Rev. X 5(2), 021001 (2015)

    Google Scholar 

  12. Vazquez, H., Skouta, R., Schneebeli, S., Kamenetska, M., Breslow, R., Venkataraman, L.: Probing the conductance superposition law in single-molecule circuits with parallel paths. Nat. Nanotechnol. 7(10), 663 (2012)

    Article  ADS  Google Scholar 

  13. Wacker, A., Karlström, O., Linke, H., Karlström, G.: Increasing thermoelectric performance using coherent transport. Phys. Rev. B 84(11), 113415 (2011)

    Article  ADS  Google Scholar 

  14. Misra, A., Singh, U., Bhattacharya, S., Pati, A.K.: Energy cost of creating quantum coherence. Phys. Rev. A 93(5), 052335 (2016)

    Article  ADS  Google Scholar 

  15. Aberg, J.: Catalytic coherence. Phys. Rev. Lett. 113(15), 150402 (2014)

    Article  ADS  Google Scholar 

  16. Narasimhachar, V., Gour, G.: Low-temperature thermodynamics with quantum coherence. Nat. Commun. 6, 7689 (2015)

    Article  ADS  Google Scholar 

  17. Ćwikliński, P., Studziński, M., Horodecki, M., Oppenheim, J.: Limitations for thermodynamical processing of coherences. Phys. Rev. Lett. 115(21), 210403 (2014)

    Article  Google Scholar 

  18. Lloyd, S.: Quantum coherence in biological systems. J. Phys. Conf. Ser. 302, 012037 (2011)

    Article  Google Scholar 

  19. Li, C.M., Lambert, N., Chen, Y.N., Chen, G.Y., Nori, F.: Witnessing quantum coherence: from solid-state to biological systems. Sci. Rep. 2(11), 885 (2012)

    Article  Google Scholar 

  20. Huelga, S.F., Plenio, M.B.: Vibrations, quanta and biology. Contemp. Phys. 54(4), 181–207 (2013)

    Article  ADS  Google Scholar 

  21. Levi, F., Mintert, F.: A quantitative theory of coherent delocalization. New J. Phys. 16(3), 935–941 (2014)

    Article  Google Scholar 

  22. Plenio, M., Huelga, S.: Dephasing assisted transport: quantum networks and biomolecules. New J. Phys. 10(11), 2952–2965 (2012)

    Google Scholar 

  23. Rebentrost, P., Mohseni, M., Aspuru-Guzik, A.: Role of quantum coherence and environmental fluctuations in chromophoric energy transport. J. Phys. Chem. B 113(29), 9942–7 (2009)

    Article  Google Scholar 

  24. 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 

  25. Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140101 (2014)

    Article  Google Scholar 

  26. Rastegin, A.E.: Relative error of state-dependent cloning. Phys. Rev. A 66(4), 519–531 (2002)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  28. 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 

  29. 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 

  30. 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 

  31. Yao, Y., Xiao, X., Ge, L., Sun, C.P.: Quantum coherence in multipartite systems. Phys. Rev. A 92(2), 022112 (2015)

    Article  ADS  Google Scholar 

  32. Xi, Z., Li, Y., Fan, H.: Quantum coherence and correlations in quantum system. Sci. Rep. 5, 10922 (2015)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  34. Yu, X.-D., Zhang, D.J., Liu, C.L., Tong, D.M.: Measure-independent freezing of quantum coherence. Phys. Rev. A 93(6), 060303 (2016)

    Article  ADS  Google Scholar 

  35. Spengler, C., Huber, M., Brierley, S., Adaktylos, T., Hiesmayr, B.C.: Entanglement detection via mutually unbiased bases. Phys. Rev. A 86(2), 8260–8269 (2012)

    Article  Google Scholar 

  36. Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191(2), 363–381 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  37. Gottesman, D.: Class of quantum error correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54(3), 1862–1868 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  38. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 78, 405 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Vaidman, L., Aharonov, Y., Albert, D.Z.: How to ascertain the values of \(\sigma _{x}, \sigma _{y}\) and \(\sigma _{z}\) of a spin-1/2 particle. Phys. Rev. Lett. 58, 1385 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  40. Englert, B.G., Aharonov, Y.: The mean king’s problem: prime degrees of freedom. Phys. Rev. A 284(1), 1–5 (2001)

    MathSciNet  Google Scholar 

  41. Durt, T., Englert, B.-G., Bengtsson, I., Zyczkowski, K.: On mutually unbiased bases. Int. J. Quantum Inf. 8(04), 535–640 (2010)

    Article  MATH  Google Scholar 

  42. Ivanovic, I.D.: An inequality for the sum of entropies of unbiased quantum measurements. J. Phys. A Gen. Phys. 25, 363–365 (1995)

    Article  Google Scholar 

  43. Sánchez, J.: Entropic uncertainty and certainty relations for complementary observables. Phys. Lett. Lett. 173, 233 (1993)

    Article  ADS  Google Scholar 

  44. Ren, L.H., Fan, H.: General fine-grained uncertainty relation and the second law of thermodynamics. Phys. Lett. A 90, 052110 (2016)

    Google Scholar 

  45. Chen, B., Fei, S.M.: Unextendible maximally entangled bases and mutually unbiased bases. Phys. Rev. A 88(3), 169–169 (2013)

    Google Scholar 

  46. Lang, M.D., Cave, C.M.: Quantum discord and the geometry of Bell-diagonal states. Phys. Rev. Lett. 105(15), 150501 (2010)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics, Tonghua Normal University, Tonghua, 134001, Jilin, China

    Yao-Kun Wang

  2. Research Center for Mathematics, College of Mathematics, Tonghua Normal University, Tonghua, 134001, Jilin, China

    Yao-Kun Wang

  3. The Branch Campus of Tonghua Normal University, Tonghua, 134001, Jilin, China

    Li-Zhu Ge

  4. Department of Mathematics, College of Sciences, Yanbian University, Yanji, 133002, China

    Yuan-Hong Tao

Authors
  1. Yao-Kun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li-Zhu Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuan-Hong Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuan-Hong Tao.

Additional information

Publisher's Note

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

Y.-H. Tao: This work is supposed by Natural Science Foundation of China under Number 11761073.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, YK., Ge, LZ. & Tao, YH. Quantum coherence in mutually unbiased bases. Quantum Inf Process 18, 164 (2019). https://doi.org/10.1007/s11128-019-2283-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-019-2283-9

Keywords

Search

Navigation