Advertisement

Nonexistence of n-qubit unextendible product bases of size \(2^n-5\)

  • Lin Chen
  • Dragomir Ž. Đoković
Article

Abstract

It is known that the n-qubit system has no unextendible product bases (UPBs) of cardinality \(2^n-1\), \(2^n-2\) and \(2^n-3\). On the other hand, the n-qubit UPBs of cardinality \(2^n-4\) exist for all \(n\ge 3\). We prove that they do not exist for cardinality \(2^n-5\).

Keywords

Unextendible product basis (UPB) Multiqubit Positive partial transpose (PPT) 

Notes

Acknowledgements

LC was supported by Beijing Natural Science Foundation (4173076), the NNSF of China (Grant No. 11501024), and the Fundamental Research Funds for the Central Universities (Grant Nos. KG12001101, ZG216S1760 and ZG226S17J6). The second author was supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada Discovery Grant 5285.

References

  1. 1.
    Alon, N., Lovsz, L.: Unextendible product bases. J. Combinatorial Theory Ser. A 95, 169–179 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Augusiak, R., Fritz, T., Kotowski, Ma., Kotowski, Mi, Pawlowski, M., Lewenstein, M., Acin, A.: Tight bell inequalities with no quantum violation from qubit unextendible product bases. Phys. Rev. A 8, 042113 (2012)ADSCrossRefGoogle Scholar
  3. 3.
    Bennett, C.H., DiVincenzo, D.P., Fuchs, C.A., Mor, T., Rains, E., Shor, P.W., Smolin, J.A., Wootters, W.K.: Quantum nonlocality without entanglement. Phys. Rev. A 59, 1070 (1999)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bennett, C.H., DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 5385 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bravyi, S.B.: Unextendible product bases and locally unconvertible bound entangled states. Quantum Inf. Process. 3, 309 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, J., Johnston, N.: The minimum size of unextendible product bases in the bipartite case (and some multipartite cases). Commun. Math. Phys. 333, 351–365 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, L., Đoković, D.Ž.: Description of rank four entangled states of two qutrits having positive partial transpose. J. Math. Phys. 52, 122203 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chen, L., Đoković, D.Ž.: Qubit-qudit states with positive partial transpose. Phys. Rev. A 86, 062332 (2012)ADSCrossRefGoogle Scholar
  9. 9.
    Chen, L., Đoković, D.Ž.: Separability problem for multipartite states of rank at most 4. J. Phys. A. Math. Theor. 46, 275304 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, L., Đoković D.Ž.: Orthogonal product bases of four qubits. J. Phys. A Math. Theor., 50: 395301 (2017)Google Scholar
  11. 11.
    DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases, uncompletable product bases and bound entanglement. Commun. Math. Phys. 238, 379–410 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Feng, K.: Unextendible product bases and 1-factorization of complete graphs. Disc. Appl. Math. 154, 942–949 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Feng, Y., Shi, Y.: Characterizing locally indistinguishable orthogonal product states. IEEE Trans. Inf. Theory 55(6), 2799–806 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Guhne, O., Seevinck, M.: Separability criteria for genuine multiparticle entanglement. New J. Phys. 12, 053002 (2010)ADSCrossRefMATHGoogle Scholar
  15. 15.
    Han, K.H., Kye, S.-H.: Construction of multi-qubit optimal genuine entanglement witnesses. J. Phys. A 49, 175303 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Johnston, N.: The minimum size of qubit unextendible product bases. In: Proceedings of the 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC) (2013)Google Scholar
  17. 17.
    Johnston, N.: The structure of qubit unextendible product bases. J. Phys. A. Math. Theor. 47, 424034 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Johnston, N.: Complete characterization of all unextendible product bases on 4 qubits (2014) www.njohnston.ca/4qubitupbs.txt
  19. 19.
    Kraus, B., Cirac, J.I., Karnas, S., Lewenstein, M.: Separability in \(2 \times N\) composite quantum systems. Phys. Rev. A 61, 062302 (2000)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Kruskal, J.B.: Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra and Appl. 18(2), 95–138 (1977)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Landsberg, J.M.: Tensors: Geometry and Applications, Amer. Math. Society, Graduate Studies in Mathematics v. 128, (2012)Google Scholar
  22. 22.
    Tura, J., Augusiak, R., Hyllus, P., Kus, M., Samsonowicz, J., Lewenstein, M.: Four-qubit entangled symmetric states with positive partial transpositions. Phys. Rev. A 85, 060302 (2012)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and Systems ScienceBeihang UniversityBeijingChina
  2. 2.International Research Institute for Multidisciplinary ScienceBeihang UniversityBeijingChina
  3. 3.Department of Pure Mathematics and Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada

Personalised recommendations