Enhancement of the Quantum Parameter Estimation in Yang-Baxter Systems

Abstract

Since Yang-Baxter systems are easy to prepare and manipulate in quantum information experiments they are of increasing interest in the estimation of physical parameters. We investigate the dynamics of quantum Fisher information for the optimal estimation of parameters using two-qubit pure and different mixed states under action of the Yang-Baxter matrices. Although quantum Fisher information is monotonically decreasing under the action of a quantum channel, we have shown that mitigation of these decreases providing relative enhancements in quantum Fisher information is possible by means of Yang-Baxter matrices which model uniter quantum channels or noises.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2

Notes

  1. 1.

    Another unitary solutions to the YBE as quantum gates comes from using extra physical parameters (rapidity parameter) that are related to statistical physics. The solutions to the YBE with the rapidity parameter allow many new unitary solutions [39].

References

  1. 1.

    Giovannetti, V., Lloyd, S., Maccone, L.: Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330–1336 (2004)

    ADS  Google Scholar 

  2. 2.

    Giovannetti, V., Lloyd, S., Maccone, L.: Quantum metrology. Phys. Rev. Lett. 96, 010401 (2006)

    ADS  MathSciNet  Google Scholar 

  3. 3.

    Giovannetti, V., Lloyd, S., Maccone, L.: Advances in quantum metrology. Nature Photon. 5, 222–229 (2011)

    ADS  Google Scholar 

  4. 4.

    Fisher, R.A.: Theory of statistical estimation. Proc. Camb. Phil. Soc. 22, 700–725 (1925)

    ADS  MATH  Google Scholar 

  5. 5.

    Cramér, H.: Mathematical Methods of Statistics. Princeton University Press, Princeton (1946)

  6. 6.

    Rao, C.R.: Information and the accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–89 (1945)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Yang, C.N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19, 1312–1315 (1967)

    ADS  MathSciNet  MATH  Google Scholar 

  8. 8.

    Yang, C.N.: S matrix for the one-dimensional N-body problem with repulsive or attractive δ,-function interaction. Phys. Rev. 168, 1920 (1968)

    ADS  Google Scholar 

  9. 9.

    Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982); Baxter, R.J.: Partition function of the Eight-Vertex lattice model. Ann. Phys. 70, 193–228 (1972)

    ADS  Google Scholar 

  10. 10.

    Drinfeld, V.G.: Hopf algebras and the quantum Yang-Baxter equation. Soviet Math. Dokl. 32, 254–258 (1985)

    Google Scholar 

  11. 11.

    Kitaev, A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  12. 12.

    Kauffman, L.H., Lomonaco, S.J. Jr: Braiding operators are universal quantum gates. New J. Phys. 36, 134 (2004)

    Google Scholar 

  13. 13.

    Zhang, Y., Kauffman, L.H., Ge, M.L.: Universal quantum gate, Yang-Baxterization and Hamiltonian. Int. J. Quant. Inf. 3, 669 (2005)

    MATH  Google Scholar 

  14. 14.

    Zhang, Y., Ge, M.L.: GHZ states, almost-complex structure and Yang-Baxter equation. Quant. Inf. Proc. 6, 363 (2007)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Zhang, Y., Rowell, E.C., Wu, Y.S., Wang, Z.H., Ge, M.L.: From extraspecial twogroups to GHZ states. arXiv:quant-ph/0706.1761(2007)

  16. 16.

    Chen, J.L., Xue, K., Ge, M.L.: Braiding transformation, entanglement swapping, and Berry phase in entanglement space. Phys. Rev. A 76, 042324 (2007)

    ADS  Google Scholar 

  17. 17.

    Chen, J.L., Xue, K., Ge, M.L.: Berry phase and quantum criticality in Yang-Baxter systems. Ann. Phys. 323, 2614 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  18. 18.

    Chen, J.L., Xue, K., Ge, M.L.: All pure two-qudit entangled states generated via a universal Yang-Baxter matrix assisted by local unitary transformations. Chin. Phys. Lett. 26, 080306 (2009)

    ADS  Google Scholar 

  19. 19.

    Brylinski, J.L., Brylinski, R.: Universal quantum gates. In: Brylinski, R., Chen, G. (eds.) Mathematics of Quantum Computation. Chapman Hall/CRC Press, Boca Raton (2002)

  20. 20.

    Wang, G., Xue, K., Wu, C., Liang, H., Oh, C.H.: Entanglement and Berry phase in a new Yang-Baxter system. J. Phys. A Math. Theor. 42, 125207 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  21. 21.

    Hu, S.-W., Xue, K., Ge, M.L.: Optical somulation of the Yang-Baxter equation. Phys. Rev. A 78, 022319 (2008)

    ADS  Google Scholar 

  22. 22.

    Hu, T., Ren, H., Xue, K.: Tripartite entanglement sudden death in Yang-Baxter systems. Quantum Inf. Process. 10, 705–715 (2011)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Yu, L.-W., Zhao, Q., Ge, M.L.: Factorized three-body S-matrix restrained by the Yang-Baxter equation and quantum entanglements. Annals of Physics 348, 106–126 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  24. 24.

    Yu, L.-W., Ge, M.L.: More about the doubling degeneracy operators associated with Majorana fermions and Yang-Baxter equation. Sci. Rep. 5, 8102 (2015)

    Google Scholar 

  25. 25.

    Helstrom, C.W.: Quantum Detection and Estimation Theory. Academic Press, New York (1976)

    Google Scholar 

  26. 26.

    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Publishing Company, Amsterdam (1982)

    Google Scholar 

  27. 27.

    Braunstein, S.L., Caves, C.M.: Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72, 3439 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  28. 28.

    Paris, M.G.A.: Quantum estimation for quantum technology. Int. J. Quantum Inform. 07, 125 (2009)

    MATH  Google Scholar 

  29. 29.

    Liu, J., Jing, X., Wang, X.: Phase-matching condition for enhancement of phase sensitivity in quantum metrology. Phys. Rev. A 88, 042316 (2013)

    ADS  Google Scholar 

  30. 30.

    Zhang, Y.M., Li, X.W., Yang, W., Jin, G.R.: Quantum Fisher information of entangled coherent states in the presence of photon loss. Phys. Rev. A 88, 043832 (2013)

    ADS  Google Scholar 

  31. 31.

    Liu, J., Jing, X.X., Zhong, W., Wang, X.G.: Quantum fisher information for density matrices with arbitrary ranks. Commun. Theor. Phys. 61, 45–50 (2014)

    ADS  MATH  Google Scholar 

  32. 32.

    Jing, X.X., Liu, J., Zhong, W., Wang, X.G.: Quantum fisher information of entangled coherent states in a lossy Mach-Zehnder interferometer. Commun. Theor. Phys. 61, 115–120 (2014)

    ADS  Google Scholar 

  33. 33.

    Liu, J., Yuan, H., Lu, X.M., Wang, X.G.: Quantum Fisher information matrix and multiparameter estimation. arXiv:1907.08037 (2019)

  34. 34.

    Boixo, S., Flammia, S.T., Caves, C.M., Geremia, J.M.: Generalized Limits for Single-Parameter Quantum Estimation. Phys. Rev. Lett. 98, 090401 (2007)

    ADS  Google Scholar 

  35. 35.

    Liu, J., Jing, X.X., Wang, X.G.: Quantum metrology with unitary parametrization processes. Sci. Rep. 5, 8565 (2015)

    Google Scholar 

  36. 36.

    Taddei, M.M., Escher, B.M., Davidovich, L., de Matos Filho, R.L.: Quantum Speed Limit for Physical Processes. Phys. Rev. Lett. 110, 050402 (2013)

    ADS  Google Scholar 

  37. 37.

    Dye, H.: Unitary solutions to the Yang-Baxter equation in dimension four. Quantum Inf. Process. 2, 117–152 (2003)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Kauffman, L.H., Lomonaco, S.J.: Braiding operators are universal quantum gates. New J. Phys. 6, 1–40 (2004)

    MathSciNet  Google Scholar 

  39. 39.

    Zhang, Y., Kauffman, L.H., Ge, M.L.: Yang-Baxterizations, universal quantum gates and Hamiltonians. Quantum Inf. Process. 4(3), 159–197 (2005)

    MathSciNet  Google Scholar 

  40. 40.

    Chen, J.L., Xue, K., Ge, M.L.: Braiding transformation, entanglement swapping, and Berry phase in entanglement space. Phys. Rev. A 76, 042324 (2007)

    ADS  Google Scholar 

  41. 41.

    Jimbo, M. (ed.): Yang-Baxter Equations in Integrable Systems. World Scientific, Singapore (1990)

  42. 42.

    Slingerland, J.K., Bais, F.A.: Quantum groups and non-abelian braiding in quantum hall systems. Nucl. Phys. B 612, 229 (2001)

    ADS  MathSciNet  MATH  Google Scholar 

  43. 43.

    Badurek, G., Rauch, H., Zeilinger, A., Bauspiess, W., Bonse, U.: Phase-shift and spin-rotation phenomena in neutron interferometry. Phys. Rev. D 14, 1177 (1976)

    ADS  Google Scholar 

  44. 44.

    Zeilinger, A.: Complementarity in neutron interferometry. Phys. B 137, 235 (1986)

    Google Scholar 

  45. 45.

    Franko, J.M., Rowell, E.C., Wang, Z.: Extraspecial 2-groups and images of braid group representations. J. Knot Theory Ramif. 15, 413–427 (2006)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Luo, S.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)

    ADS  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Durgun Duran.

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

Verify currency and authenticity via CrossMark

Cite this article

Duran, D. Enhancement of the Quantum Parameter Estimation in Yang-Baxter Systems. Int J Theor Phys 59, 2091–2100 (2020). https://doi.org/10.1007/s10773-020-04481-6

Download citation

Keywords

  • Quantum Fisher information
  • Yang-Baxter equation
  • Quantum parameter estimation