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Quantum state tomography for generic pure states

  • ShiLin Huang
  • JianXin Chen
  • YouNing Li
  • Bei ZengEmail author
Article

Abstract

We examine the problem of whether a multipartite pure quantum state can be uniquely determined by its reduced density matrices. We show that a generic pure state in three party Hilbert space HAHBHC, where dim(HA) = 2 and dim(HB) = dim(HC), can be uniquely determined by its reduced states on subsystems HAHBHC. Then, we generalize the conclusion to the case that dim(H1) > 2. As a corollary, we show that a generic N-qudit pure quantum state is uniquely determined by only two of its \(\left\lceil {\frac{{N + 1}}{2}} \right\rceil \)-particle reduced density matrices. Furthermore, our results indicate a method to uniquely determine a generic N-qudit pure state of dimension D = dN with only O(D) local measurements, which is an improvement compared to the previous known approach that uses O(Dlog2D) or O(Dlog D) local measurements.

Keywords

state reconstruction quantum tomography foundations of quantum mechanics measurement theory 

References

  1. 1.
    G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, Lecture Notes Phys. 649, 9 (2004).Google Scholar
  2. 2.
    A. I. Lvovsky, and M. G. Raymer, Rev. Mod. Phys. 81, 299 (2009).ADSCrossRefGoogle Scholar
  3. 3.
    H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, Nature 438, 643 (2005).ADSCrossRefGoogle Scholar
  4. 4.
    M. Baur, A. Fedorov, L. Steffen, S. Filipp, M. P. da Silva, and A. Wallraff, Phys. Rev. Lett. 108, 040502 (2012), arXiv: 1107.4774.ADSCrossRefGoogle Scholar
  5. 5.
    G. M. D’Ariano, M. D. Laurentis, M. G. A. Paris, A. Porzio, and S. Solimeno, J. Opt. B-Quantum Semiclass. Opt. 4, S127 (2002).CrossRefGoogle Scholar
  6. 6.
    D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland, Nature 438, 639 (2005).ADSCrossRefGoogle Scholar
  7. 7.
    M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, Nat. Commun. 4, 2295 (2013), arXiv: 1211.7036.ADSCrossRefGoogle Scholar
  8. 8.
    S. T. Flammia, D. Gross, Y. K. Liu, and J. Eisert, New J. Phys. 14, 095022 (2012), arXiv: 1205.2300.ADSCrossRefGoogle Scholar
  9. 9.
    D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Phys. Rev. Lett. 105, 150401 (2010), arXiv: 0909.3304.ADSCrossRefGoogle Scholar
  10. 10.
    D. Lu, T. Xin, N. Yu, Z. Ji, J. Chen, G. Long, J. Baugh, X. Peng, B. Zeng, and R. Laflamme, Phys. Rev. Lett. 116, 230501 (2016), arXiv: 1511.00581.ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, Nat. Commun. 1, 149 (2010), arXiv: 1101.4366.ADSCrossRefGoogle Scholar
  12. 12.
    C. H. Baldwin, I. H. Deutsch, and A. Kalev, Phys. Rev. A 93, 052105 (2016), arXiv: 1605.02109.ADSCrossRefGoogle Scholar
  13. 13.
    H. Kosaka, T. Inagaki, Y. Rikitake, H. Imamura, Y. Mitsumori, and K. Edamatsu, Nature 457, 702 (2009).ADSCrossRefGoogle Scholar
  14. 14.
    H. Y. Wang, W. Q. Zheng, N. K. Yu, K. R. Li, D. W. Lu, T. Xin, C. Li, Z. F. Ji, D. Kribs, B. Zeng, X. H. Peng, and J. F. Du, Sci. China-Phys. Mech. Astron. 59, 100313 (2016), arXiv: 1604.06277.CrossRefGoogle Scholar
  15. 15.
    J. Y. Chen, Z. Ji, Z. X. Liu, X. Qi, N. Yu, B. Zeng, and D. Zhou, Sci. China-Phys. Mech. Astron. 60, 020311 (2017), arXiv: 1605.06357.ADSCrossRefGoogle Scholar
  16. 16.
    J. Chen, C. Guo, Z. Ji, Y. T. Poon, N. Yu, B. Zeng, and J. Zhou, Sci. China-Phys. Mech. Astron. 60, 020312 (2017), arXiv: 1606.07422.ADSCrossRefGoogle Scholar
  17. 17.
    N. Linden, S. Popescu, and W. K. Wootters, Phys. Rev. Lett. 89, 207901 (2002).ADSCrossRefGoogle Scholar
  18. 18.
    N. Linden, and W. K. Wootters, Phys. Rev. Lett. 89, 277906 (2002).ADSCrossRefGoogle Scholar
  19. 19.
    L. Di´osi, Phys. Rev. A 70, 010302 (2004).ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    N. S. Jones, and N. Linden, Phys. Rev. A 71, 012324 (2005).ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    J. Chen, Z. Ji, M. B. Ruskai, B. Zeng, and D. L. Zhou, J. Math. Phys. 53, 072203 (2012), arXiv: 1205.3682.ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    J. Chen, Z. Ji, B. Zeng, and D. L. Zhou, Phys. Rev. A 86, 022339 (2012), arXiv: 1110.6583.ADSCrossRefGoogle Scholar
  23. 23.
    J. Chen, H. Dawkins, Z. Ji, N. Johnston, D. Kribs, F. Shultz, and B. Zeng, Phys. Rev. A 88, 012109 (2013), arXiv: 1212.3503.ADSCrossRefGoogle Scholar
  24. 24.
    E. Schmidt, Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten (Berlin, Springer-Verlag, 1989), pp. 190–233.Google Scholar
  25. 25.
    A. Acín, A. Andrianov, L. Costa, E. Jané, J. I. Latorre, and R. Tarrach, Phys. Rev. Lett. 85, 1560 (2000).ADSCrossRefGoogle Scholar
  26. 26.
    H. A. Carteret, A. Higuchi, and A. Sudbery, J. Math. Phys. 41, 7932 (2000).ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    J. M. Lee, Introduction to Smooth Manifolds (Springer, New York, 2003), p. 127.CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • ShiLin Huang
    • 1
  • JianXin Chen
    • 2
  • YouNing Li
    • 3
  • Bei Zeng
    • 4
    • 5
    Email author
  1. 1.Institute for Interdisciplinary Information SciencesTsinghua UniversityBeijingChina
  2. 2.Aliyun Quantum LaboratoryHangzhouChina
  3. 3.Department of Physics and Collaborative Innovation Center of Quantum MatterTsinghua UniversityBeijingChina
  4. 4.Department of Mathematics & StatisticsUniversity of GuelphGuelphCanada
  5. 5.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada

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