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Stronger uncertainty relations of mixed states

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Abstract

The Heisenberg–Robertson uncertainty relation bounds the product of the variances in the two possible measurement outcomes in terms of the expectation of the commutator of the observables. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two noncompatible observables. Here, we give two stronger uncertainty relations, relating to the sum of variances with respect to density matrix, whose lower bounds are guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system; moreover, two stronger uncertainty relations in terms of the product of the variances of two observables are established. Also, several stronger uncertainty relations for three observables are established, relating to the sum and product of variances with respect to density matrix, respectively.

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References

  1. Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172–198 (1927)

    Article  ADS  Google Scholar 

  2. Pati, A.K., Sahu, P.K.: Sum uncertainty in quantum theorem. Phys. lett. A 367, 177–181 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  3. Huang, Y.: Variance-based uncertainty relations. Phys. Rev. A 86, 024101 (2012)

    Article  ADS  Google Scholar 

  4. Rivas, A., Luis, A.: Characterization of quantum angular-momentum fluctuations via principal components. Phys. Rev. A 77, 022105 (2008)

    Article  ADS  Google Scholar 

  5. Chen, B., Fei, S.M.: Sum uncertainty relations for arbitrary \(N\) incompatible observables. Sci. Rep. 5, 14238 (2015)

    Article  ADS  Google Scholar 

  6. Yao, Y., Xiao, X., Wang, X., Sun, C.P.: Implications and applications of the variance-based uncertainty equalities. Phys. Rev. A 91, 062113 (2015)

    Article  ADS  Google Scholar 

  7. Zheng, X., Ma, S.Q., Zhang, G.F., Fan, H., Liu, W.M.: Unifed and exact framework for variance-based uncertainty relations. Sci. Rep. 10, 150 (2020)

    Article  ADS  Google Scholar 

  8. 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  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  13. Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  14. Gühne, O., Tóth, G.: Entanglement detection. Phys. Rep. 474(1–6), 1–75 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  15. Wehner, S., Winter, A.: Entropic uncertainty relations-a survey. New J. Phys. 12, 025009 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  16. Candès, E., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    Article  MathSciNet  Google Scholar 

  17. Zidan, M., Abdel-Aty, A.-H., Younes, A., Zanaty, E.A., El-khayat, I., Abdel-Aty, M.: A novel algorithm based on entanglement measurement for improving speed of quantum algorithms. Appl. Math. Inf. Sci. 12(1), 265–269 (2018)

    Article  MathSciNet  Google Scholar 

  18. Zidan, M., Abdel-Aty, A.-H., Nguyene, D.M., Mohamed, A.S.A., Al-Sbou, Y., Eleuch, H., Abdel-Aty, M.: A quantum algorithm based on entanglement measure for classifying multivariate function into novel hidden classes. Results Phys. 15, 102549 (2019)

    Article  Google Scholar 

  19. Zidan, M., Abdel-Aty, A.-H., El-shafei, M., Feraig, M., Al-Sbou, Y., Eleuch, H., Abdel-Aty, M.: Quantum classification algorithm based on competitive learning neural network and entanglement measure. Appl. Sci. 9(7), 1277 (2019)

    Article  Google Scholar 

  20. Abdel-Aty, A.-H., Kadry, H., Zidan, M., Al-Sbou, Y., Zanaty, E.A., Abdel-Aty, M.: A quantum classification algorithm for classification incomplete patterns based on entanglement measure. J. Intell. Fuzzy Syst. 38(3), 2809–2816 (2020)

    Article  Google Scholar 

  21. Schrödinger, E.: About Heisenberg uncertainty relation. In: Proceedings of The Prussian Academy of Sciences, Physics-Mathematical Section XIX, vol. 293 (1930)

  22. Maccone, L., Pati, A.K.: Strong uncertaionty relations for all incompatible observables. Phys. Rev. Lett. 113, 260401 (2014)

    Article  ADS  Google Scholar 

  23. Hofmann, H.F., Takeuchi, S.: Violation of local uncertainty relation as a signature of entanglement. Phys. Rev. A 68, 032103 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  24. Holevo, A.S.: A generalization of the Rao-Cramér inequality. Probab. Appl. 18, 359–362 (1974)

    Article  Google Scholar 

  25. Song, Q.C., Qiao, C.F.: Stronger Schrödinger-like uncertainty relations. Phys. Lett. A 380(37), 2925–2930 (2016)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This subject was supported by the NNSF of China (Nos. 11701011, 11761001, 11761003), the NSF of Ningxia (No. 2020AAC03229), and the First-Class Disciplines Foundation of Ningxia (No.NXYLXK2017B09).

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Correspondence to Yajing Fan.

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Fan, Y., Cao, H., Chen, L. et al. Stronger uncertainty relations of mixed states. Quantum Inf Process 19, 256 (2020). https://doi.org/10.1007/s11128-020-02761-y

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