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

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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.

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References

  1. Alon N., Lovász L.: Unextendible product bases. J. Comb. Theory Ser. A 95, 169–179 (2001)

    Article  MATH  Google Scholar 

  2. 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)

    Article  ADS  Google Scholar 

  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, 1070–1091 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  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, 5385–5388 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  5. Bhat B.: A completely entangled subspace of maximal dimension. Int. J. Quantum Inf. 4, 325–330 (2006)

    Article  MATH  Google Scholar 

  6. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Feng K.: Unextendible product bases and 1-factorization of complete graphs. Discrete Appl. Math. 154, 942–949 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Herschkowitz D., Schneider H.: Ranks of zero patterns and sign patterns. Linear Multilinear Algebra 34, 3–19 (1993)

    Article  Google Scholar 

  9. Johnston, N.: Code for computing some unextendible product bases. http://www.njohnston.ca/publications/minimum-upbs/code/ (2012)

  10. 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. 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)

    Article  ADS  Google Scholar 

  12. Pedersen, T.B.: Characteristics of unextendible product bases. Master’s thesis, Aarhus Universitet, Datalogisk Institut (2002)

  13. Skowronek Ł.: Three-by-three bound entanglement with general unextendible product bases. J. Math. Phys. 52, 122202 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  14. Terhal B.M.: A family of indecomposable positive linear maps based on entangled quantum states. Linear Algebra Appl. 323, 61–73 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada

    Jianxin Chen

  2. Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Jianxin Chen & Nathaniel Johnston

  3. UTS-AMSS Joint Research Laboratory for Quantum Computation and Quantum Information Processing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Jianxin Chen

Authors
  1. Jianxin Chen
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  2. Nathaniel Johnston
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Correspondence to Nathaniel Johnston.

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Communicated by A. Winter

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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). https://doi.org/10.1007/s00220-014-2186-7

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  • DOI: https://doi.org/10.1007/s00220-014-2186-7

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