[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Computing Geometric Measure of Entanglement for Symmetric Pure States via the Jacobian SDP Relaxation Technique

  • Published:
Journal of the Operations Research Society of China Aims and scope Submit manuscript

Abstract

The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue (US-eigenvalue) for symmetric complex tensors, which can be taken as a multilinear optimization problem in complex number field. In this paper, we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem. Then we use Jacobian semidefinite relaxation method to solve it. Some numerical examples are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Lathauwer, L.D., Moor, B.D., Vandewalle, J.: On the best rank-1 and rank-(\(r_1, r_2,., r_n\)) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Zhang, T., Golub, G.H.: Rank-one approximation to high order tensors. SIAM J. Matrix Anal. Appl. 23, 534–550 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23, 863–884 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ni, G., Wang, Y.: On the best rank-1 approximation to higher-order symmetric tensors. Math. Comput. Model. 46, 1345–1352 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: Proceeding of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, vol. 1, pp. 129–132. IEEE Computer Society Press, Piscataway (2005)

  8. Ni, G., Qi, L., Bai, M.: Geometric measure of entanglement and U-eigenvalues of tensors. SIAM J. Matrix Anal. Appl. 35, 73–87 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  9. Shimony, A.: Degree of entanglement. Ann. N. Y. Acad. Sci. 755, 675–679 (1995)

    Article  MathSciNet  Google Scholar 

  10. Wei, T.C., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003)

    Article  Google Scholar 

  11. Hubener, R., Kleinmann, M., Wei, T.C., Guillen, C.G., Guhne, O.: Geometric measure of entanglement for symmetric states. Phys. Rev. A 80, 032324 (2009)

    Article  MathSciNet  Google Scholar 

  12. Hu, S., Qi, L., Zhang, G.: Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors. Phys. Rev. A 93, 012304 (2016)

    Article  MathSciNet  Google Scholar 

  13. Ni, G., Bai, M.: Spherical optimization with complex variables for computing US-eigenpairs. Comput. Optim. Appl. (2016). doi:10.1007/s10589-016-9848-7

  14. Qi, L., Wang, F., Wang, Y.: Z-eigenvalue methods for a global polynomial optimization problem. Math. Program. 118, 301–316 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hu, S., Huang, Z.H., Qi, L.: Finding the extreme Z-eigenvalues of tensors via a sequential semidefinite programming method. Numer. Linear Algebra Appl. 20, 972–984 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32, 1095–1124 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cui, C., Dai, Y., Nie, J.: All real eigenvalues of symmetric tensors. SIAM J. Matrix Anal. Appl. 35, 1582–1601 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhang, X., Ling, C., Qi, L.: The best rank-1 approximation of a symmetric tensor and related spherical optimization problems. SIAM J. Matrix Anal. Appl. 33, 806–821 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Han, L.: An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numer. Algebra Control Optim. 3, 583–599 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Nie, J.: An exact Jacobian SDP relaxation for polynomial optimization. Math. Program. 137, 225–255 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Nie, J.: The hierarchy of local minimums in polynomial optimization. Math. Program. 151(2), 555–583 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Henrion, D., Lasserre, J.B., Loefberg, J.: Gloptipoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24, 761–779 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sturm, J.F.: Sedumi 1.02: A MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11, 625–653 (1999)

Download references

Acknowledgments

The authors thank the two anonymous referees for their very useful comments.

Author information

Authors and Affiliations

  1. College of Science, National University of Defense Technology, Changsha, 410073, China

    Bing Hua, Gu-Yan Ni & Meng-Shi Zhang

Authors
  1. Bing Hua
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gu-Yan Ni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Meng-Shi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gu-Yan Ni.

Additional information

This work was supported by the Research Programme of National University of Defense Technology (No. ZK16-03-45).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hua, B., Ni, GY. & Zhang, MS. Computing Geometric Measure of Entanglement for Symmetric Pure States via the Jacobian SDP Relaxation Technique. J. Oper. Res. Soc. China 5, 111–121 (2017). https://doi.org/10.1007/s40305-016-0135-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40305-016-0135-1

Keywords

Mathematics Subject Classification

Search

Navigation