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

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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.

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References

  1. Wigner, E.P., Yanase, M.M.: Information contents of distributions. Proc. Natl. Acad. Sci. USA 49, 910–918 (1963)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Luo, S., Zhang, Q.: On skew information. IEEE Trans. Inf. Theory 50, 1778–1782 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Luo, S.: Heisenberg uncertainty relation for mixed states. Phys. Rev. A 72, 042110 (2005)

    Article  ADS  Google Scholar 

  4. Furuichi, S., Yanagi, K., Kuriyama, K.: Trace inequalities on a generalized Wigner–Yanase skew information. J. Math. Anal. Appl. 356, 179–185 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Furuichi, S.: Schrödinger uncertainty relation with Wigner–Yanase skew information. Phys. Rev. A 82, 034101 (2010)

    Article  ADS  Google Scholar 

  6. Yanagi, K.: Uncertainty relation on Wigner–Yanase–Dyson skew information. J. Math. Anal. Appl. 365, 12–18 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Yanagi, K.: Wigner–Yanase–Dyson skew information and uncertainty relation. J. Phys. Conf. Ser. 201, 012015 (2010)

    Article  MATH  Google Scholar 

  8. Ko, C.K., Yoo, H.J.: Uncertainty relation associated with a monotone pair skew information. J. Math. Anal. Appl. 383, 208–214 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Müller, M., Rotter, I.: Phase lapses in open quantum systems and the non-Hermitian Hamilton operator. Phys. Rev. A 80(4), 042705 (2009)

    Article  ADS  Google Scholar 

  16. Rotter, I.: A non-Hermitian Hamilton operator and the physics of open quantum systems. J. Phys. A Math. Theor. 42, 153001 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. Long, G.L.: General quantum interference principle and duality computer. Commun. Theor. Phys. 45, 825 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  21. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  23. Moiseyev, N.: Non-Hermitian Quantum Mechanics. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  24. Matzkin, A.: Weak measurements in non-Hermitian systems. J. Phys. A Math. Theor. 45, 444023 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Pati, A.K., Singh, U., Sinha, U.: Measuring non-Hermitian operators via weak values. Phys. Rev. A 92, 052120 (2015)

    Article  ADS  Google Scholar 

  26. Rastegin, A.E.: Entropic uncertainty relations and quasi-Hermitian operators. J. Phys. A Math. Theor. 45, 444026 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dou, Y.N., Du, H.K.: Generalizations of the Heisenberg and Schrödinger uncertainty relations. J. Math. Phys. 54, 103508 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Dou, Y.N., Du, H.K.: Note on the Wigner–Yanase–Dyson skew information. Int. J. Theor. Phys. 53, 952–958 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Gudder, S.: Operator probability theory. Int. J. Pure Appl. Math. 39, 511–525 (2007)

    MathSciNet  MATH  Google Scholar 

  34. Bender, C.M., Brody, D.C., Jones, H.F., Meister, B.K.: Faster than Hermitian quantum mechanics. Phys. Rev. Lett. 98, 040403 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China

    Yajing Fan

  2. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, China

    Huaixin Cao

  3. School of Ethnic Education, Shaanxi Normal University, Xi’an, 710062, China

    Wenhua Wang

  4. Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin, 300071, China

    Huixian Meng

  5. Department of Mathematics, Changji College, Changji, 831100, China

    Liang Chen

Authors
  1. Yajing Fan
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  2. Huaixin Cao
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  3. Wenhua Wang
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  4. Huixian Meng
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  5. Liang Chen
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Correspondence to Yajing Fan.

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Fan, Y., Cao, H., Wang, W. et al. Uncertainty relations with the generalized Wigner–Yanase–Dyson skew information. Quantum Inf Process 17, 157 (2018). https://doi.org/10.1007/s11128-018-1906-x

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  • DOI: https://doi.org/10.1007/s11128-018-1906-x

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