Skip to main content
SpringerLink
Log in

Criteria for SLOCC and LU Equivalence of Generic Multi-qudit States

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In this paper, we study the stochastic local operation and classical communication (SLOCC) and local unitary (LU) equivalence for multi-qudit states by mode-n matricization of the coefficient tensors. We establish a new scheme of using the CANDECOMP/PARAFAC (CP) decomposition of tensors to find necessary and sufficient conditions between the mode-n unfolding and SLOCC&LU equivalence for pure multi-qudit states. For multipartite mixed states, we present a necessary and sufficient condition for LU equivalence and necessary condition for SLOCC equivalence.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  3. Bennett, C. H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)

    Article  ADS  MATH  Google Scholar 

  4. Boschi, D., Branca, S., De Martini, F., Hardy, L., Popescu, S.: Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 80, 1121 (1998)

    Article  ADS  MATH  Google Scholar 

  5. Albeverio, S., Fei, S.M.: Teleportation of general finite-dimensional quantum systems. Phys. Lett. A 276, 8–11 (2000)

    Article  ADS  MATH  Google Scholar 

  6. Albeverio, S., Fei, S.M., Yang, W.L.: Optimal teleportation based on Bell measurements. Phys. Rev. A 66, 012301 (2002)

    Article  ADS  Google Scholar 

  7. Deutsch, D., Ekert, A., Jozsa, R., Macchiavello, C., Popescu, S., Sanpera, A.: Quantum privacy amplification and the security of quantum cryptography over noisy channels. Phys. Rev. Lett. 77, 2818–2821 (1996)

    Article  ADS  Google Scholar 

  8. Griffiths, R.B., Niu, C.-S.: Optimal eavesdropping in quantum cryptography. II. A quantum circuit. Phys. Rev. A 56, 1173–1176 (1997)

    Article  ADS  Google Scholar 

  9. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. R. Soc. London Ser. A 439, 553–558 (1992)

    Article  ADS  MATH  Google Scholar 

  10. Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys Rev. Lett. 79, 325 (1997)

    Article  ADS  Google Scholar 

  11. Ekert, A., Jozsa, R.: Quantum computation and Shor’s factoring algorithm. Rev. Mod. Phys. 68, 733 (1996)

    Article  ADS  Google Scholar 

  12. Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881 (1992)

    Article  ADS  MATH  Google Scholar 

  13. Bennett, C.H., Popescu, S., Rohrlich, D., Smolin, J.A., Thapliyal, A.V.: Exact and asymptotic measures of multipartite pure-state entanglement. Phys Rev. A 63, 012307 (2001)

    Article  ADS  Google Scholar 

  14. Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)

    Article  ADS  Google Scholar 

  15. Verstraete, F., Dehaene, J., De Moor, B., Verschelde, H.: Four qubits can be entangled in nine different ways. Phys Rev. A 65, 052112 (2002)

    Article  ADS  Google Scholar 

  16. Acín, A., Bruß, D., Lewenstein, M., Sanpera, A.: Classification of mixed three-qubit states. Phys Rev. Lett. 87, 040401 (2001)

    Article  ADS  Google Scholar 

  17. Li, X.R., Li, D.F.: Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states. Phys Rev. A 86, 042332 (2012)

    Article  ADS  Google Scholar 

  18. Wang, S.H., Lu, Y., Long, G.L.: Entanglement classification of 2 × 2 × 2 × d quantum systems via the ranks of the multiple coefficient matrices. Phys Rev. A 87, 062305 (2013)

    Article  ADS  Google Scholar 

  19. Wang, S.H., Lu, Y., Gao, M., Cui, J.L., Li, J.L.: Classification of arbitrary-dimensional multipartite pure states under stochastic local operations and classical communication using the rank of coefficient matrix. J. Phys A Math. Theor. 46, 105303 (2013)

    Article  ADS  MATH  Google Scholar 

  20. Holweck, F., Luque, J.G., Thibon, J.Y.: Entanglement of four-qubit systems: a geometric atlas with polynomial compass II (the tame world). J. Math. Phys. 58, 022201 (2017)

    Article  ADS  MATH  Google Scholar 

  21. Turner, J.: A new degree bound for local unitary and n-qubit SLOCC invariants, arXiv:1706.00634 (2017)

  22. Macia̧żek, T., Sawicki, A.: Asymptotic properties of entanglement polytopes for large number of qubits, vol. 51 (2018)

  23. Zangi, S. M., Li, J. L., Qiao, C.F.: Entanglement classification of four-partite states under the SLOCC. J. Phys. A Math. Theor. 50, 325301 (2017)

    Article  ADS  MATH  Google Scholar 

  24. Shi, X.: The stabilizer for n-qubit symmetric states. Chin Phys. B 27, 100311 (2018)

    Article  ADS  Google Scholar 

  25. Huang, Y., Yu, H.P., Miao, F., Han, T.Y., Zhang, X.J.: Mathematical framework for describing multipartite entanglement in terms of rows or columns of coefficient matrices. Int. J. Quant. Inf. 20, 2150035 (2022)

    Article  MATH  Google Scholar 

  26. Fei, S.M., Jing, N.: Equivalence of quantum states under local unitary transformations. Phys. Lett. A 342, 77–81 (2005)

    Article  ADS  MATH  Google Scholar 

  27. Kraus, B.: Local unitary equivalence of multipartite pure states. Phys Rev. Lett. 104, 020504 (2010)

    Article  ADS  MATH  Google Scholar 

  28. Liu, B., Li, J.L., Li, X.K., Qiao, C.F.: Local unitary classification of arbitrary dimensional multipartite pure states. Phys Rev. Lett. 108, 050501 (2012)

    Article  ADS  Google Scholar 

  29. Verstraete, F., Dehaene, J., De Moor, B.: Normal forms and entanglement measures for multipartite quantum states. Phys Rev. A 68, 012103 (2003)

    Article  ADS  Google Scholar 

  30. Zhang, T.G., Zhao, M.J., Li, M., Fei, S.M., Li-Jost, X.: Criterion of local unitary equivalence for multipartite states. Phys Rev. A 88, 042304 (2013)

    Article  ADS  Google Scholar 

  31. Li, M., Zhang, T.G., Fei, S.M., Li-Jost, X., Jing, N.: Local unitary equivalence of multi-qubit mixed quantum states. Phys. Rev. A 89, 062325 (2014)

    Article  ADS  Google Scholar 

  32. Makhlin, Y.: Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations. Quant Inf. Process. 1, 243–252 (2002)

    Article  Google Scholar 

  33. Turner, J., Morton, J.: A complete set of invariants for LU-equivalence of density operators. SIGMA, pp. 13 (2017)

  34. Jing, N., Li, M., Li-Jost, X., Zhang, T.G., Fei, S.M.: SLOCC invariants for multipartite mixed states. J. Phys. A Math. Theor. 47, 215303 (2014)

    Article  ADS  MATH  Google Scholar 

  35. Jing, N., Fei, S.M., Li, M., Li-Jost, X., Zhang, T.G.: Local unitary invariants of generic multiqubit states. Phys Rev. A 92, 022306 (2015)

    Article  ADS  Google Scholar 

  36. Sun, B.Z., Fei, S.M., Wang, Z.X.: On local unitary equivalence of two and three-qubit states. Sci Rep. 7, 4869 (2017)

    Article  ADS  Google Scholar 

  37. Kruskal, J.B.: Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Alg. Appl. 18, 95–138 (1977)

    Article  MATH  Google Scholar 

  38. Hitchcock, F.L.: The expression of a tensor or a polyadic as a sum of products. J. Math. Phys. 6, 164–189 (1927)

    Article  MATH  Google Scholar 

  39. Qi, L.Q., Luo, Z.Y.: Tensor analysis: spectral theory and special tensors. Philadelphia: Society for Industrial and Applied Mathematics (2017)

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

    Article  ADS  MATH  Google Scholar 

  41. Smilde, A., Bro, R., Geladi, P.: Multi-Way Analysis: Applications in the Chemical Sciences. Wiley, England (2004)

    Book  Google Scholar 

  42. Horn, R. A., Johnson, C.R.: Topics in matrix analysis. Cambridge England: Cambridge university (1991)

  43. Chang, J.M., Jing, N.: Local unitary equivalence of generic multi-qubits based on the CP decomposition. Int. J. Theor. Phys. 61, 137 (2022)

    Article  MATH  Google Scholar 

  44. Chen, K., Wu, L.A.: A matrix realignment method for recognizing entanglement. Quant. Inf Comp. 3, 193–202 (2003)

    MATH  Google Scholar 

  45. Sun, L.L., Li, J.L., Qiao, C.F.: Classification of the entangled states of 2 × L × M × N. Quant. Inf Process. 14, 229–245 (2015)

    Article  ADS  MATH  Google Scholar 

  46. Zhang, T.G., Zhao, M.J., Huang, X.F.: Criterion for SLOCC equivalence of multipartite quantum states. J. Phys. A Math. Theor. 49, 405301 (2016)

    Article  MATH  Google Scholar 

  47. Van Loan, C.F.: The ubiquitous Kronecker product. J. Comput. Appl. Math. 123, 85–100 (2000)

    Article  ADS  MATH  Google Scholar 

Download references

Acknowledgements

The research is partially supported by Simons Foundation grant no. 523868 and National Natural Science Foundation of China under grant nos. 12126351, 12126314 and 11861031. This project is also supported by the specific research fund of the Innovation Platform for Academicians of Hainan Province under Grant No.YSPTZX202215 and Hainan Provincial Natural Science Foundation of China under Grant No.121RC539.

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, China

    Jingmei Chang

  2. Department of Mathematics, North Carolina State University, Raleigh, USA

    Naihuan Jing

  3. School of Mathematics and Statistics, Hainan Normal University, Haikou, China

    Tinggui Zhang

Authors
  1. Jingmei Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Naihuan Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tinggui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Naihuan Jing.

Ethics declarations

All data generated during the study are included in the article.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chang, J., Jing, N. & Zhang, T. Criteria for SLOCC and LU Equivalence of Generic Multi-qudit States. Int J Theor Phys 62, 6 (2023). https://doi.org/10.1007/s10773-022-05267-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10773-022-05267-8

Keywords

Search

Navigation