Uncertainty relations with the generalized Wigner–Yanase–Dyson skew information

  • Yajing Fan
  • Huaixin Cao
  • Wenhua Wang
  • Huixian Meng
  • Liang Chen
Article
  • 36 Downloads

Abstract

The uncertainty principle in quantum mechanics is a fundamental relation with different forms, including Heisenberg’s uncertainty relation and Schrödinger’s uncertainty relation. We introduce the generalized Wigner–Yanase–Dyson correlation and the related quantities. Various properties of them are discussed. Finally, we establish several generalizations of uncertainty relation expressed in terms of the generalized Wigner–Yanase–Dyson skew information.

Keywords

Uncertainty relation Generalized Wigner–Yanase–Dyson correlation Generalized Wigner–Yanase–Dyson skew information 

Notes

Acknowledgements

This subject was supported by the SRP for the Ningxia Universities (No. NGY2017156).

References

  1. 1.
    Wigner, E.P., Yanase, M.M.: Information contents of distributions. Proc. Natl. Acad. Sci. USA 49, 910–918 (1963)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Luo, S., Zhang, Q.: On skew information. IEEE Trans. Inf. Theory 50, 1778–1782 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Luo, S.: Heisenberg uncertainty relation for mixed states. Phys. Rev. A 72, 042110 (2005)ADSCrossRefGoogle Scholar
  4. 4.
    Furuichi, S., Yanagi, K., Kuriyama, K.: Trace inequalities on a generalized Wigner–Yanase skew information. J. Math. Anal. Appl. 356, 179–185 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Furuichi, S.: Schrödinger uncertainty relation with Wigner–Yanase skew information. Phys. Rev. A 82, 034101 (2010)ADSCrossRefGoogle Scholar
  6. 6.
    Yanagi, K.: Uncertainty relation on Wigner–Yanase–Dyson skew information. J. Math. Anal. Appl. 365, 12–18 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Yanagi, K.: Wigner–Yanase–Dyson skew information and uncertainty relation. J. Phys. Conf. Ser. 201, 012015 (2010)CrossRefMATHGoogle Scholar
  8. 8.
    Ko, C.K., Yoo, H.J.: Uncertainty relation associated with a monotone pair skew information. J. Math. Anal. Appl. 383, 208–214 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Furuichi, S., Yanagi, K.: Schrödinger uncertainty relation, Wigner–Yanase–Dyson skew information and metric adjusted correlation measure. J. Math. Anal. Appl. 388, 1147–1156 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, B., Fei, S.M., Long, G.L.: Sum uncertainty relations based on Wigner–Yanase skew information. Quantum Inf. Process. 15(6), 2639–2648 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chen, B., Cao, N.P., Fei, S.M., Long, G.L.: Variance-based uncertainty relations for incompatible observables. Quantum Inf. Process. 15, 3909–3917 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cheng, W.W., Du, Z.Z., Gong, L.Y., Zhao, S.M., Liu, J.M.: Signature of topological quantum phase transitions via Wigner–Yanase skew information. Europhys. Lett. 108, 46003 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    Guo, J.L., Wei, J.L., Qin, W., Mu, Q.X.: Examining quantum correlations in the XY spin chain by local quantum uncertainty. Quantum Inf. Process. 14, 1429–1442 (2015)ADSCrossRefMATHGoogle Scholar
  14. 14.
    Cheng, W.W., Zhang, Z.J., Gong, L.Y., Zhao, S.M.: Universal role of quantum uncertainty in Anderson metal–insulator transition. Ann. Phys. 370, 67 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Müller, M., Rotter, I.: Phase lapses in open quantum systems and the non-Hermitian Hamilton operator. Phys. Rev. A 80(4), 042705 (2009)ADSCrossRefGoogle Scholar
  16. 16.
    Rotter, I.: A non-Hermitian Hamilton operator and the physics of open quantum systems. J. Phys. A Math. Theor. 42, 153001 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Alber, G., Delgado, A., Gisin, N., Jex, I.: Generalized quantum XOR-gate for quantum teleportation and state purification in arbitrary dimensional Hilbert spaces. Quantum Phys. arXiv:quant-ph/0008022v1 (2000)
  18. 18.
    Long, G.L.: General quantum interference principle and duality computer. Commun. Theor. Phys. 45, 825 (2006)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Guo, Z.H., Cao, H.X., Chen, Z.L., Yin, J.C.: Operational properties and matrix representations of quantum measures. Chin. Sci. Bull. 56, 1671 (2011)CrossRefGoogle Scholar
  20. 20.
    Guo, Z.H., Cao, H.X.: Existence and construction of a quantum channel with given inputs and outputs. Chin. Sci. Bull. 57, 4346–4350 (2012)CrossRefGoogle Scholar
  21. 21.
    Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Moiseyev, N.: Non-Hermitian Quantum Mechanics. Cambridge University Press, Cambridge (2011)CrossRefMATHGoogle Scholar
  24. 24.
    Matzkin, A.: Weak measurements in non-Hermitian systems. J. Phys. A Math. Theor. 45, 444023 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pati, A.K., Singh, U., Sinha, U.: Measuring non-Hermitian operators via weak values. Phys. Rev. A 92, 052120 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    Rastegin, A.E.: Entropic uncertainty relations and quasi-Hermitian operators. J. Phys. A Math. Theor. 45, 444026 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Dey, S., Fring, A., Khantoul, B.: Hermitian versus non-Hermitian representations for minimal length uncertainty relations. J. Phys. A Math. Theor. 46, 335304 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Dou, Y.N., Du, H.K.: Generalizations of the Heisenberg and Schrödinger uncertainty relations. J. Math. Phys. 54, 103508 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Dou, Y.N., Du, H.K.: Note on the Wigner–Yanase–Dyson skew information. Int. J. Theor. Phys. 53, 952–958 (2014)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Li, Q., Cao, H.X., Du, H.K.: A generalization of Schrödinger’s uncertainty relation described by the Wigner–Yanase skew information. Quantum Inf. Process. 14, 1513–1522 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Chen, Z.L., Liang, L.L., Li, H.J., Wang, W.H.: A generalized uncertainty relation. Int. J. Theor. Phys. 54, 2644–2651 (2015)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Chen, Z.L., Liang, L.L., Li, H.J., Wang, W.H.: Two generalized Wigner–Yanase skew information and their uncertainty relations. Quantum Inf. Process. 15, 5107–5118 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Gudder, S.: Operator probability theory. Int. J. Pure Appl. Math. 39, 511–525 (2007)MathSciNetMATHGoogle Scholar
  34. 34.
    Bender, C.M., Brody, D.C., Jones, H.F., Meister, B.K.: Faster than Hermitian quantum mechanics. Phys. Rev. Lett. 98, 040403 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceNorth Minzu UniversityYinchuanChina
  2. 2.School of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina
  3. 3.School of Ethnic EducationShaanxi Normal UniversityXi’anChina
  4. 4.Theoretical Physics Division, Chern Institute of MathematicsNankai UniversityTianjinChina
  5. 5.Department of MathematicsChangji CollegeChangjiChina

Personalised recommendations