Communications in Mathematical Physics

, Volume 333, Issue 1, pp 351–365 | Cite as

The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)

Article
First Online:
Received:
Accepted:

Abstract

A long-standing open question asks for the minimum number of vectors needed to form an unextendible product basis in a given bipartite or multipartite Hilbert space. A partial solution was found by Alon and Lovász (J. Comb. Theory Ser. A, 95:169–179, 2001), but since then only a few other cases have been solved. We solve all remaining bipartite cases, as well as a large family of multipartite cases.

Keywords

Minimum Size Orthogonality Condition Permutation Matrix Positive Partial Transpose Nonzero Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AL01.
    Alon N., Lovász L.: Unextendible product bases. J. Comb. Theory Ser. A 95, 169–179 (2001)CrossRefMATHGoogle Scholar
  2. ASH+11.
    Augusiak, R., Stasinska, J., Hadley, C., Korbicz, J.K., Lewenstein, M., Ac쬬 A.: Bell inequalities with no quantum violation and unextendible product bases. Phys. Rev. Lett. 107, 070401 (2011)ADSCrossRefGoogle Scholar
  3. BDF+99.
    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, 1070–1091 (1999)ADSCrossRefMathSciNetGoogle Scholar
  4. BDM+99.
    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, 5385–5388 (1999)ADSCrossRefMathSciNetGoogle Scholar
  5. Bha06.
    Bhat B.: A completely entangled subspace of maximal dimension. Int. J. Quantum Inf. 4, 325–330 (2006)CrossRefMATHGoogle Scholar
  6. DMS+03.
    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)ADSCrossRefMathSciNetMATHGoogle Scholar
  7. Fen06.
    Feng K.: Unextendible product bases and 1-factorization of complete graphs. Discrete Appl. Math. 154, 942–949 (2006)CrossRefMathSciNetMATHGoogle Scholar
  8. HS93.
    Herschkowitz D., Schneider H.: Ranks of zero patterns and sign patterns. Linear Multilinear Algebra 34, 3–19 (1993)CrossRefGoogle Scholar
  9. Joh12.
    Johnston, N.: Code for computing some unextendible product bases. http://www.njohnston.ca/publications/minimum-upbs/code/ (2012)
  10. Joh13.
    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). doi: 10.4230/LIPIcs.TQC.2013.93
  11. LSM11.
    Leinaas J.M., Sollid P.Ø., Myrheim J.: Unextendible product bases and extremal density matrices with positive partial transpose. Phys. Rev. A 84, 042325 (2011)ADSCrossRefGoogle Scholar
  12. Ped02.
    Pedersen, T.B.: Characteristics of unextendible product bases. Master’s thesis, Aarhus Universitet, Datalogisk Institut (2002)Google Scholar
  13. Sko11.
    Skowronek Ł.: Three-by-three bound entanglement with general unextendible product bases. J. Math. Phys. 52, 122202 (2011)ADSCrossRefMathSciNetGoogle Scholar
  14. Ter01.
    Terhal B.M.: A family of indecomposable positive linear maps based on entangled quantum states. Linear Algebra Appl. 323, 61–73 (2001)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of GuelphGuelphCanada
  2. 2.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  3. 3.UTS-AMSS Joint Research Laboratory for Quantum Computation and Quantum Information ProcessingAcademy of Mathematics and Systems Science, Chinese Academy of SciencesBeijingChina

Personalised recommendations