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State-independent uncertainty relations and entanglement detection

  • Chen Qian
  • Jun-Li Li
  • Cong-Feng Qiao
Article
  • 155 Downloads

Abstract

The uncertainty relation is one of the key ingredients of quantum theory. Despite the great efforts devoted to this subject, most of the variance-based uncertainty relations are state-dependent and suffering from the triviality problem of zero lower bounds. Here we develop a method to get uncertainty relations with state-independent lower bounds. The method works by exploring the eigenvalues of a Hermitian matrix composed by Bloch vectors of incompatible observables and is applicable for both pure and mixed states and for arbitrary number of N-dimensional observables. The uncertainty relation for the incompatible observables can be explained by geometric relations related to the parallel postulate and the inequalities in Horn’s conjecture on Hermitian matrix sum. Practical entanglement criteria are also presented based on the derived uncertainty relations.

Keywords

Uncertainty relations Entanglement detection Bloch vector Horn’s inequalities 

Notes

Acknowledgements

This work was supported in part by the Ministry of Science and Technology of the People’s Republic of China(2015CB856703); by the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No.XDB23030100; and by the National Natural Science Foundation of China(NSFC) under the Grants 11375200 and 11635009.

References

  1. 1.
    Busch, P., Heinonen, T., Lahti, P.J.: Heisenberg’s uncertainty principle. Phys. Rep. 452, 155 (2007)ADSCrossRefGoogle Scholar
  2. 2.
    Hofmann, H.F., Takeuchi, S.: Violation of local uncertainty relations as a signature of entanglement. Phys. Rev. A. 68, 032103 (2003)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Gühne, O.: Characterizing entanglement via uncertainty relations. Phys. Rev. Lett. 92, 117903 (2004)ADSCrossRefGoogle Scholar
  4. 4.
    Fuchs, C.A., Peres, A.: Quantum-state disturbance versus information gain: uncertainty relations for quantum information. Phys. Rev. A. 53, 2038 (1996)ADSCrossRefGoogle Scholar
  5. 5.
    Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172 (1927)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik 44, 326 (1927)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163 (1929)ADSCrossRefGoogle Scholar
  8. 8.
    Schrödinger, E.: Situngsberichte der Preussischen Akademie der Wissenschaften. Physikalisch-mathematische. Klasse. 14, 296 (1930)Google Scholar
  9. 9.
    Maccone, L., Pati, A.K.: Stronger uncertainty relations for all incompatible observables. Phys. Rev. Lett. 113, 260401 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    Song, Q.-C., Qiao, C.-F.: Stronger Schrödinger-like uncertainty relations. Phys. Lett. A. 380, 2925 (2016)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Song, Q.-C., Qiao, C.-F.: Uncertainty equalities and uncertainty relation in weak measurement. arXiv:1505.02233 (2015)
  12. 12.
    Chen, B., Fei, S.-M.: Sum uncertainty relations for arbitrary N incompatible observables. Sci. Rep. 5, 14238 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Bagchi, S., Pati, A.K.: Uncertainty relations for general unitary operators. Phys. Rev. A. 94, 042104 (2016)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Xiao, Y.-L., Jing, N.-H., Li-Jost, X.-Q., Fei, S.-M.: Weighted uncertainty relations. Sci. Rep. 6, 23201 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    Chen, B., Cao, N.-P., Fei, S.-M., Long, G.-L.: Variance-based uncertainty relations for incompatible observables. Quantum Inf. Process. 15, 3909 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Song, Q.-C., Li, J.-L., Peng, G.-X., Qiao, C.-F.: A stronger multi-observable uncertainty relation. Sci. Rep. 7, 44764 (2017)ADSCrossRefGoogle Scholar
  17. 17.
    Qin, H.-H., Fei, S.-M., Li-Jost, X.-Q.: Multi-observable uncertainty relations in product form of variances. Sci. Rep. 6, 31192 (2016)ADSCrossRefGoogle Scholar
  18. 18.
    Chen, B., Fei, S.-M., Long, G.-L.: Sum uncertainty relations based on Wigner–Yanase skew information. Quantum Inf. Process. 15, 2639 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mondal, D., Bagchi, S., Pati, A.K.: Tighter uncertainty and reverse uncertainty relations. Phys. Rev. A. 95, 052117 (2017)ADSCrossRefGoogle Scholar
  20. 20.
    Zhang, J., Zhang, Y., Yu, C.-S.: Stronger uncertainty relations with improvable upper and lower bounds. Quantum Inf. Process. 16, 131 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Park, Y.M.: Improvement of uncertainty relations for mixed states. J. Math. Phys. 46, 042109 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Li, J.-L., Qiao, C.-F.: Reformulating the quantum uncertainty relation. Sci. Rep. 5, 12708 (2015)ADSCrossRefGoogle Scholar
  23. 23.
    Abbott, A.A., Alzieu, P., Hall, M.J.W., Branciard, C.: Tight state-independent uncertainty relations for qubits. Mathematics 4, 8 (2016)CrossRefzbMATHGoogle Scholar
  24. 24.
    Schwonnek, R., Dammeier, L., Werner, R.F.: State-independent uncertainty relations and entanglement detection in noisy systems. Phys. Rev. Lett. 119, 170404 (2017)ADSCrossRefGoogle Scholar
  25. 25.
    Dammeier, L., Schwonnek, R., Werner, R.F.: Uncertainty relations for angular momentum. New J. Phys. 17, 093046 (2015)ADSCrossRefGoogle Scholar
  26. 26.
    Horn, A.: Eigenvalues of sums of Hermitian matrices. Pac. J. Math. 12, 225 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Fulton, W.: Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Am. Math. Soc. 37, 209 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hioe, F.T., Eberly, J.H.: \(N\)-Level coherence vector and higher conservation laws in quantum optics and quantum mechanics. Phys. Rev. Lett. 47, 838 (1981)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Kimura, G.: The Bloch vector for \(N\)-level systems. Phys. Lett. A 314, 339 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Byrd, M.S., Khaneja, N.: Characterization of the positivity of the density matrix in terms of the coherence vector representation. Phys. Rev. A. 68, 062322 (2003)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Hofmann, H.F.: Uncertainty characteristics of generalized quantum measurements. Phys. Rev. A. 67, 022106 (2003)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Li, J.-L., Qiao, C.-F.: A necessary and sufficient criterion for the separability of quantum state. Sci. Rep. 8, 1442 (2018)ADSCrossRefGoogle Scholar
  33. 33.
    de Vicente, J.I.: Separability criteria based on the Bloch representation of density matrices. Quantum Inf. Comput. 7, 624 (2007)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Rudolph, O.: Further results on the cross norm criterion for separability. Quantum Inf. Process. 4, 219 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Chen, K., Wu, L.-A.: A matrix realignment method for recognizing entanglement. Quantum Inf. Comput. 3, 193 (2003)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Gühne, O., Hyllus, P., Gittsovich, O., Eiert, J.: Covariance matrices and the separablity problem. Phys. Rev. Lett. 99, 130504 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Li, J.-L., Qiao, C.-F.: Separable decompositions of bipartite mixed states. Quantum Inf. Process. arXiv: 1708.05336 (2017)
  38. 38.
    Jevtic, S., Pusey, M., Jennings, D., Rudolph, T.: Quantum steering ellipsoids. Phys. Rev. Lett. 113, 020402 (2014)ADSCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of Chinese Academy of SciencesBeijingChina
  2. 2.Key Laboratory of Vacuum PhysicsUniversity of Chinese Academy of SciencesBeijingChina

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