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Uncertainty regions of observables and state-independent uncertainty relations

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Abstract

The optimal state-independent lower bounds for the sum of variances or deviations of observables are of significance for the growing number of experiments that reach the uncertainty limited regime. We present a framework for computing the tight uncertainty relations of variance or deviation via determining the uncertainty regions, which are formed by the tuples of two or more of quantum observables in random quantum states induced from the uniform Haar measure on the purified states. From the analytical formulae of these uncertainty regions, we present state-independent uncertainty inequalities satisfied by the sum of variances or deviations of two, three and arbitrary many observables, from which experimentally friend entanglement detection criteria are derived for bipartite and tripartite systems.

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References

  1. Heisenberg, W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik 43, 172 (1927). https://doi.org/10.1007/BF01397280

    Article  MATH  ADS  Google Scholar 

  2. Dammeier, L., Schwonneck, R., Werner, R.F.: Uncertainty relations for angular momentum. New J. Phys. 17, 093046 (2015). https://doi.org/10.1088/1367-2630/17/9/093046

    Article  MATH  ADS  Google Scholar 

  3. Li, J.L., Qiao, C.F.: Reformulating the quantum uncertainty relation. Sci. Rep. 5, 12708 (2015). https://doi.org/10.1038/srep12708

    Article  ADS  Google Scholar 

  4. de Guise, H., Maccone, L., Sanders, B.C., Shukla, N.: State-independent uncertainty relations. Phys. Rev. A 98, 042121 (2018). https://doi.org/10.1103/PhysRevA.98.042121

    Article  ADS  Google Scholar 

  5. Giorda, P., Maccone, L., Riccardi, A.: State-independent uncertainty relations from eigenvalue minimization. Phys. Rev. A 99, 052121 (2019). https://doi.org/10.1103/PhysRevA.99.052121

    Article  MathSciNet  ADS  Google Scholar 

  6. Xiao, Y., Guo, C., Meng, F., Jing, N., Yung, M.-H.: Incompatibility of observables as state-independent bound of uncertainty relations. Phys. Rev. A 100, 032118 (2019). https://doi.org/10.1103/PhysRevA.100.032118

    Article  MathSciNet  ADS  Google Scholar 

  7. Sponar, S., Danner, A., Obigane, K., Hack, S., Hasegawa, Y.: Experimental test of tight state-independent preparation uncertainty relations for qubits. Phys. Rev. A 102, 042204 (2020). https://doi.org/10.1103/PhysRevA.102.042204

    Article  ADS  Google Scholar 

  8. Seife, C.: Do deeper principles underlie quantum uncertainty and nonlocality? Science 309, 98 (2005). https://doi.org/10.1126/science.309.5731.98

    Article  Google Scholar 

  9. Hofmann, H.F., Takeuchi, S.: Violation of local uncertainty relations as a signature of entanglement. Phys. Rev. A 68, 032103 (2003). https://doi.org/10.1103/PhysRevA.68.032103

    Article  MathSciNet  ADS  Google Scholar 

  10. Gühne, O.: Characterizing entanglement via uncertainty relations. Phys. Rev. Lett. 92, 117903 (2004). https://doi.org/10.1103/PhysRevLett.92.117903

    Article  ADS  Google Scholar 

  11. Gühne, O., Tóth, G.: Entanglement detection. Phys. Rep. 474, 1 (2009). https://doi.org/10.1016/j.physrep.2009.02.004

    Article  MathSciNet  ADS  Google Scholar 

  12. Schwonnek, R., Dammeier, L., Werner, R.F.: State-independent uncertainty relations and entanglement detection in noisy systems. Phys. Rev. Lett. 119, 170404 (2017). https://doi.org/10.1103/PhysRevLett.119.170404

    Article  ADS  Google Scholar 

  13. Qian, C., Li, J.-L., Qiao, C.-F.: State-independent uncertainty relations and entanglement detection. Quant. Inf. Process. 17, 84 (2018). https://doi.org/10.1007/s11128-018-1855-4

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. Zhao, Y.-Y., Xiang, G.Y., Hu, X.M., Liu, B.H., Li, C.F., Guo, G.C., Schwonnek, R., Wolf, R.: Entanglement detection by violations of noisy uncertainty relations: a proof of principle. Phys. Rev. Lett. 122, 220401 (2019). https://doi.org/10.1103/PhysRevLett.122.220401

    Article  ADS  Google Scholar 

  15. Oppenheim, J., Wehner, S.: The uncertainty principle determines the nonlocality of quantum mechanics. Science 330, 1072–1074 (2010). https://doi.org/10.1126/science.1192065

    Article  MathSciNet  MATH  ADS  Google Scholar 

  16. Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik 44, 326 (1927). https://doi.org/10.1007/BF01391200

    Article  MATH  ADS  Google Scholar 

  17. Weyl, H.: Gruppentheorie und Quantenmechanik. Hirzel, Leipzig (1928)

    MATH  Google Scholar 

  18. Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163 (1929). https://doi.org/10.1103/PhysRev.34.163

    Article  ADS  Google Scholar 

  19. Schrödinger, E.: Zum Heisenbergschen Unscharfeprinzip. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse 14, 296 (1930)

    MATH  Google Scholar 

  20. Busch, P., Reardon-Smith, O.: On quantum uncertainty relations and uncertainty regions. arXiv:1901.03695

  21. Vasudevrao, S., Sudha, I.R., Usha Devi, A.R., Rajagopal, A.K.: Sum uncertainty relations: uncertainty regions for qubits and qutrits. arXiv: 2101.11545v1

  22. Lubkin, E.: Entropy of an \(n\)-system from its correlation with a \(k\)-reservoir. J. Math. Phys. 19, 1028 (1978). https://doi.org/10.1063/1.523763

    Article  MATH  Google Scholar 

  23. Page, D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993). https://doi.org/10.1103/PhysRevLett.71.1291

    Article  MathSciNet  MATH  ADS  Google Scholar 

  24. Foong, S.K., Kanno, S.: Proof of Page’s conjecture on the average entropy of a subsystem. Phys. Rev. Lett. 72, 1148 (1994). https://doi.org/10.1103/PhysRevLett.72.1148

  25. Sen, S.: Average entropy of a quantum subsystem. Phys. Rev. Lett. 77, 1 (1996). https://doi.org/10.1103/PhysRevLett.71.1291

    Article  ADS  Google Scholar 

  26. Hastings, M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys. 5, 255 (2009). https://doi.org/10.1038/nphys1224

    Article  Google Scholar 

  27. Christandl, M., Doran, B., Kousidis, S., Walter, M.: Eigenvalue distributions of reduced density matrices. Commun. Math. Phys. 332, 1 (2014). https://doi.org/10.1007/s00220-014-2144-4

    Article  MathSciNet  MATH  ADS  Google Scholar 

  28. Dartois, S., Lionni, L., Nechita, I.: The joint distribution of the marginals of multipartite random quantum states. Random Matrices Theory Appl. 9, 2050010 (2020). https://doi.org/10.1142/s2010326320500100

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhang, L., Wang, J., Chen, Z.H.: Spectral density of mixtures of random density matrices for qubits. Phys. Lett. A 382, 1516 (2018). https://doi.org/10.1016/j.physleta.2018.04.018

    Article  MathSciNet  MATH  ADS  Google Scholar 

  30. Zhang, L., Jiang, Y.X., Wu, J.D.: Duistermaat–Heckman measure and the mixture of quantum states. J. Phys. A Math. Theor. 52, 495203 (2019). https://doi.org/10.1088/1751-8121/ab5297

    Article  MathSciNet  Google Scholar 

  31. Venuti, L.C., Zanardi, P.: Probability density of quantum expectation values. Phys. Lett. A 377, 1854 (2013). https://doi.org/10.1016/j.physleta.2013.05.041

    Article  MathSciNet  MATH  ADS  Google Scholar 

  32. Itzykson, C., Züber, J.B.: The planar approximation. II. J. Math. Phys. 21, 411 (1980). https://doi.org/10.1063/1.524438

    Article  MathSciNet  MATH  ADS  Google Scholar 

  33. Życzkowski, K., Sommers, H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A Math. Theor. 34, 7111–7125 (2001). https://doi.org/10.1088/0305-4470/34/35/335

    Article  MathSciNet  MATH  ADS  Google Scholar 

  34. Zhang, L.: Dirac delta function of matrix argument. Int. J. Theor. Phys. (2020). https://doi.org/10.1007/s10773-020-04598-8

  35. Zhang, L., Xiang, H., Li-Jost, X., Fei, S.M.: Incompatibility probability of random quantum measurements. Phys. Rev. E 100, 062139 (2019). https://doi.org/10.1103/PhysRevE.100.062139

    Article  ADS  Google Scholar 

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Acknowledgements

This work is supported by the NSF of China under Grant Nos. 11971140, 61771174 and 12075159, Beijing Natural Science Foundation (Z190005), Key Project of Beijing Municipal Commission of Education (KZ201810028042), the Academician Innovation Platform of Hainan Province, Academy for Multidisciplinary Studies, Capital Normal University, and Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology (Grant No. SIQSE202001).

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Authors and Affiliations

  1. Institute of Mathematics, Hangzhou Dianzi University, Hangzhou, 310018, People’s Republic of China

    Lin Zhang

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Shunlong Luo

  3. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

    Shao-Ming Fei

  4. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Junde Wu

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  1. Lin Zhang
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Correspondence to Lin Zhang.

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Zhang, L., Luo, S., Fei, SM. et al. Uncertainty regions of observables and state-independent uncertainty relations. Quantum Inf Process 20, 357 (2021). https://doi.org/10.1007/s11128-021-03303-w

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