Skip to main content
SpringerLink
Log in

Ordering states with Tsallis relative \(\alpha \)-entropies of coherence

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this paper, we study the ordering states with Tsallis relative \(\alpha \)-entropies of coherence and \(l_{1}\) norm of coherence for single-qubit states. Firstly, we show that any Tsallis relative \(\alpha \)-entropies of coherence and \(l_{1}\) norm of coherence give the same ordering for single-qubit pure states. However, they do not generate the same ordering for some high-dimensional states, even though these states are pure. Secondly, we also consider three special Tsallis relative \(\alpha \)-entropies of coherence for \(\alpha =2, 1, \frac{1}{2}\) and show these three measures and \(l_{1}\) norm of coherence will not generate the same ordering for some single-qubit mixed states. Nevertheless, they may generate the same ordering if we only consider a special subset of single-qubit mixed states. Furthermore, we find that any two of these three special measures generate different ordering for single-qubit mixed states. Finally, we discuss the degree of violation of between \(l_{1}\) norm of coherence and Tsallis relative \(\alpha \)-entropies of coherence. In a sense, this degree can measure the difference between these two coherence measures in ordering states.

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. Scully, M.O., Zubairy, M.S.: Quantum Optics. Can- brudge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  2. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge Univ. Press, Cambridge (2000)

    MATH  Google Scholar 

  3. Rodríguez-Rosario, C.A., Frauenheim, T., Aspuru-Guzik, A.: Thermodynamics of quantum coherence. arXiv:1308.1245

  4. Å berg, J.: Catalytic coherence. Phys. Rev. Lett. 113, 150402 (2014)

    Article  Google Scholar 

  5. Horodecki, M., Oppenheim, J.: Resource theory of quantum states out of thermal equilibrium. Nat. Commun. 4, 2059 (2013)

    Article  ADS  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

  8. Å berg, J.: Quantifying superposition. arXiv:quant-ph/0612146

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  13. 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 

  14. Napoli, C., Bromley, T.R., Cianciaruso, 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. Zhang, Y.-R., Shao, L.-H., Li, Y., Fan, H.: Quantifying coherence in infinite-dimensional systems. Phys. Rev. A 93, 012334 (2016)

    Article  ADS  Google Scholar 

  16. Yu, X.-D., Zhang, D.-J., Xu, G. F., Tong, D.M.: An alternative framework for quantifying coherence. arXiv:1606.03181

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

    Article  ADS  Google Scholar 

  18. Liu, C.L., Yu, X.D., Xu, G.F., Tong, D.M.: Ordering states with coherence measures. Quantum Inf. Process. (2016). doi:10.1007/s11128-016-1398-5

    MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  21. Cheng, S., Hall, M.J.W.: Complementarity relations for quantum coherence. Phys. Rev. A 92, 042101 (2015)

    Article  ADS  Google Scholar 

  22. Bera, M.N., Qureshi, T., Siddiqui, M.A., Pati, A.K.: Duality of quantum coherence and path distinguishability. Phys. Rev. A 92, 012118 (2015)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  26. Xu, J.: Quantifying coherence of Gaussian states. Phys. Rev. A 93, 032111 (2016)

    Article  ADS  Google Scholar 

  27. Yadin, B., Ma, J., Girolami, D., Gu, M., Vedral, V.: Quantum processes which do not use coherence. arXiv:1512.02085

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

    Article  ADS  Google Scholar 

  29. Chitambar, E., Gour, G.: Are incoherent operations physically consistent?—A critical examination of incoherent operations. arXiv:1602.06969

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  32. Eisert, J., Plenio, M.B.: A comparison of entanglement measures. J. Mod. Opt. 46, 145 (1999)

    Article  ADS  Google Scholar 

  33. Virmani, S., Plenio, M.B.: Ordering states with entanglement measures. Phys. Lett. A 268, 31 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Zyczkowski, K., Bengtsson, I.: Relativity of pure states entanglement. Ann. Phys. (NY) 295, 115 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Miranowicz, A., Grudka, A.: Ordering two-qubit states with concurrence and negativity. Phys. Rev. A 70, 032326 (2004)

    Article  ADS  Google Scholar 

  36. Wei, T.C., Nemoto, K., Goldbart, P.M., Kwiat, P.G., Munro, W.J., Verstraete, F.: Maximal entanglement versus entropy for mixed quantum states. Phys. Rev. A 67, 022110 (2003)

    Article  ADS  Google Scholar 

  37. Ziman, M., Bŭzek, V.: Entanglement-induced state ordering under local operations. Phys. Rev. A 73, 012312 (2006)

    Article  ADS  Google Scholar 

  38. Sen De, A., Sen, U.: Can there be quantum correlations in a mixture of two separable states? J. Mod. Opt. 50, 981 (2003)

    ADS  MATH  Google Scholar 

  39. Ali, M., Rau, A.R.P., Alber, G.: Quantum discord for two-qubit X states. Phys. Rev. A 81, 042105 (2010)

    Article  ADS  Google Scholar 

  40. Lang, M.D., Caves, C.M., Shaji, A.: Entropic measures of non-classical correlations. Int. J. Quantum Inf. 09, 1553 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Galve, F., Plastina, F., Paris, M.G.A., et al.: Discording power of quantum evolutions Phys. Rev. lett. 110, 010501 (2013)

    Article  ADS  Google Scholar 

  42. Okrasa, M., Walczak, Z.: On two-qubit states ordering with quantum discords. Europhys. Lett. 98, 40003 (2012)

    Article  ADS  Google Scholar 

  43. Furuichi, S., Yanagi, K., Kuriyama, K.: Fundamental properties of Tsallis relative entropy. J. Math. Phys. 45, 4868 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Hiai, F., Mosonyi, M., Petz, D., Bény, C.: Reversibility conditions for quantum operations. Rev. Math. Phy. 23, 691 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Luo, Y., Tian, T., Shao, L.-H., Li, Y.: General monogamy of Tsallis-q entropy entanglement in multiqubit systems. Phys. Rev. A 93, 062340 (2016)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This paper is supported by National Natural Science Foundation of China (Grants Nos. 11271237, 61228305, 61602291), The Higher School Doctoral Subject Foundation of Ministry of Education of China (Grant No. 20130202110001) and Fundamental Research Funds for the Central Universities (No. 2016CBY003).

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, China

    Fu-Gang Zhang & Yongming Li

  2. College of Computer Science, Shaanxi Normal University, Xi’an, 710119, China

    Lian-He Shao, Yu Luo & Yongming Li

  3. College of Mathematics and Statistics, Huangshan University, Huangshan, 245041, China

    Fu-Gang Zhang

Authors
  1. Fu-Gang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lian-He Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yongming Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongming Li.

Appendix

Appendix

We provide a proof of \(\frac{\partial r}{\partial z}\ge 0\). The first equation comes from the derivation of \(r_{\frac{1}{2}}\) with respect to z. In the second equation, we use distributive law and then merge similar items. The last inequality comes from the fact \(2-z^2-t^2\ge 2\sqrt{1-\sqrt{z^2+t^2}}\).

$$\begin{aligned} \frac{\partial r_{\frac{1}{2}}}{\partial z}= & {} \frac{1}{\sqrt{2}}\left[ \frac{\sqrt{1+\sqrt{z^2+t^2}}\left( \sqrt{z^2+t^2}+z\right) }{\sqrt{z^2+t^2}}\right. \\&\left. +\frac{\sqrt{1-\sqrt{z^2+t^2}} \left( \sqrt{z^2+t^2}-z\right) }{\sqrt{z^2+t^2}}\right] \\&\left[ \frac{\sqrt{2}}{8}\frac{z\left( \sqrt{z^2+t^2}+z\right) }{\sqrt{1+ \sqrt{z^2+t^2}(z^2+t^2)}}- \frac{\sqrt{2}}{8}\frac{z\left( \sqrt{z^2+t^2}-z\right) }{\sqrt{1- \sqrt{z^2+t^2}(z^2+t^2)}}\right. \\&\left. -\frac{1}{2\sqrt{2}}\frac{\sqrt{1+\sqrt{z^2+t^2}} z\left( \sqrt{z^2+t^2}+z\right) }{(z^2+t^2)^\frac{3}{2}}\right. \\&\left. +\frac{1}{2\sqrt{2}} \frac{\sqrt{1+\sqrt{z^2+t^2}} \left( \sqrt{z^2+t^2}+z\right) }{(z^2+t^2)}\right. \\&\left. -\frac{1}{2\sqrt{2}} \frac{\sqrt{1-\sqrt{z^2+t^2}}z\left( \sqrt{z^2+t^2}-z\right) }{ (z^2+t^2)^\frac{3}{2}}\right. \\&\left. -\frac{1}{2\sqrt{2}} \frac{\sqrt{1-\sqrt{z^2+t^2}} \left( \sqrt{z^2+t^2}-z\right) }{(z^2+t^2)}\right] \\ \end{aligned}$$
$$\begin{aligned}&+\frac{1}{\sqrt{2}}\left[ \frac{\sqrt{1+\sqrt{z^2+t^2}} \left( \sqrt{z^2+t^2}-z\right) }{\sqrt{z^2+t^2}}\right. \\&\left. +\frac{\sqrt{1-\sqrt{z^2+t^2}} \left( \sqrt{z^2+t^2}+z\right) }{\sqrt{z^2+t^2}}\right] \\&\left[ \frac{\sqrt{2}}{8}\frac{z \left( \sqrt{z^2+t^2}-z\right) }{\sqrt{1+\sqrt{z^2+t^2}(z^2+t^2)}}\right. \\&\left. - \frac{\sqrt{2}}{8}\frac{z\left( \sqrt{z^2+t^2}+z\right) }{ \sqrt{1-\sqrt{z^2+t^2}(z^2+t^2)}}\right. \\&\left. -\frac{1}{2\sqrt{2}} \frac{\sqrt{1+\sqrt{z^2+t^2}} \left( \sqrt{z^2+t^2}-z\right) }{(z^2+t^2)}\right. \\&\left. -\frac{1}{2\sqrt{2}}\frac{\sqrt{1+\sqrt{z^2+t^2}}z \left( \sqrt{z^2+t^2}-z\right) }{(z^2+t^2)^\frac{3}{2}}\right. \\&\left. +\frac{1}{2\sqrt{2}} \frac{\sqrt{1-\sqrt{z^2+t^2}} \left( \sqrt{z^2+t^2}+z\right) }{(z^2+t^2)}\right. \\&\left. -\frac{1}{2\sqrt{2}} \frac{\sqrt{1-\sqrt{z^2+t^2}}z \left( \sqrt{z^2+t^2}+z\right) }{(z^2+t^2)^\frac{3}{2}}\right] . \end{aligned}$$
$$\begin{aligned}= & {} \frac{1}{2}\frac{\left( \sqrt{z^2+t^2}+z\right) ^2}{(z^2+t^2)^\frac{3}{2}} -\frac{1}{2}\frac{\left( \sqrt{z^2+t^2}-z\right) ^2}{(z^2+t^2)^\frac{3}{2}}\\&-\frac{1}{2}\frac{z\left( \sqrt{z^2+t^2}-z\right) ^2}{(z^2+t^2)^2} -\frac{1}{2}\frac{z\left( \sqrt{z^2+t^2}+z\right) ^2}{(z^2+t^2)^2}\\&-\frac{zt^2\sqrt{1-(z^2+t^2)}}{(z^2+t^2)^2} +\frac{1}{4}\frac{zt^2\sqrt{1-\sqrt{z^2+t^2}}}{\sqrt{ \left( 1+\sqrt{z^2+t^2}\right) (z^2+t^2)^\frac{3}{2}}}\\&-\frac{1}{4}\frac{zt^2\sqrt{1+\sqrt{z^2+t^2}}}{\sqrt{1-\sqrt{z^2+t^2}}(z^2+t^2)^\frac{3}{2}}\\= & {} \frac{2z\sqrt{z^2+t^2}}{(z^2+t^2)^\frac{3}{2}}-\frac{z(2z^2+t^2)}{(z^2+t^2)^2} -\frac{zt^2\sqrt{1-(z^2+t^2)}}{(z^2+t^2)^2}\\&-\frac{1}{2}\frac{zt^2\sqrt{z^2+t^2}}{\sqrt{1-(z^2+t^2)}(z^2+t^2)^\frac{3}{2}}\\= & {} \frac{z^2}{2\sqrt{1-(z^2+t^2)}(z^2+t^2)} \left[ -2+z^2+t^2+2\sqrt{1-\sqrt{z^2+t^2}}\right] \le 0. \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, FG., Shao, LH., Luo, Y. et al. Ordering states with Tsallis relative \(\alpha \)-entropies of coherence. Quantum Inf Process 16, 31 (2017). https://doi.org/10.1007/s11128-016-1488-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-016-1488-4

Keywords

Search

Navigation