Connecting unextendible maximally entangled base with partial Hadamard matrices

  • Yan-Ling Wang
  • Mao-Sheng Li
  • Shao-Ming Fei
  • Zhu-Jun Zheng
Article
  • 144 Downloads

Abstract

We study the unextendible maximally entangled bases (UMEB) in \(\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}\) and connect the problem to the partial Hadamard matrices. We show that for a given special UMEB in \(\mathbb {C}^{d}\bigotimes \mathbb {C}^{d}\), there is a partial Hadamard matrix which cannot be extended to a Hadamard matrix in \(\mathbb {C}^{d}\). As a corollary, any \((d-1)\times d\) partial Hadamard matrix can be extended to a Hadamard matrix, which answers a conjecture about \(d=5\). We obtain that for any d there is a UMEB except for \(d=p\ \text {or}\ 2p\), where \(p\equiv 3\mod 4\) and p is a prime. The existence of different kinds of constructions of UMEBs in \(\mathbb {C}^{nd}\bigotimes \mathbb {C}^{nd}\) for any \(n\in \mathbb {N}\) and \(d=3\times 5 \times 7\) is also discussed.

Keywords

Unextendible maximally entangled bases Partial Hadamard matrices Maximally entangled state 

Notes

Acknowledgements

This work is supported by the NSFC 11475178, NSFC 11571119 and NSFC 11275131.

References

  1. 1.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  2. 2.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Horodecki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Horodecki, M.: Entanglement measures. Quantum Inf. Comput. 1, 3 (2001)MathSciNetMATHGoogle Scholar
  5. 5.
    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 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    De Rinaldis, S.: Distinguishability of complete and unextendible product bases. Phys. Rev. A 70, 022309 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bravyi, S., Smolin, J.A.: Unextendible maximally entangled bases. Phys. Rev. A 84, 042306 (2011)ADSCrossRefGoogle Scholar
  9. 9.
    Chen, B., Fei, S.M.: Unextendible maximally entangled bases and mutually unbiased bases. Phys. Rev. A 88, 034301 (2013)ADSCrossRefGoogle Scholar
  10. 10.
    Li, M.-S., Wang, Y.-L., Zheng, Z.-J.: Unextendible maximally entangled bases in \(\mathbb{C}^{d}\bigotimes \mathbb{C}^{d^{\prime }}\). Phys. Rev. A 89, 062313 (2014)ADSCrossRefGoogle Scholar
  11. 11.
    Wang, Y.-L., Li, M.-S., Fei, S.M.: Unextendible maximally entangled bases in \(\mathbb{C}^{d}\bigotimes \mathbb{C}^{d}\). Phys. Rev. A 90, 034301 (2014)ADSCrossRefGoogle Scholar
  12. 12.
    Butson, A.T.: Generalized Hadamard matrices. Proc. Am. Math. Soc. 13, 894C898 (1962)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Zauner, G.: Quantendesigns: Grundzauge einer nichtkommutativen Designtheorie (German) [Quantumdesigns: the foundations of a noncommutative design theory]. Ph.D. Thesis, Universitat Wien. http://www.mat.univie.ac.at/neum/ms/zauner (1999)
  14. 14.
    Tadej, W., Zyczkowski, K.: A concise guide to complex Hadamard matrices. Open Syst. Inf. Dyn. 13, 133C177 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Verheiden, E.: Integral and rational completions of combinatorial matrices. J. Comb. Theory Ser. A 25, 267 (1978)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    de Launey, W., Levin, D.A.: A Fourier-analytic approach to counting partial Hadamard matrices. Cryptogr. Commun. 2, 307 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Banica, T., Skalski, A.: The quantum algebra of partial Hadamard matrices. Linear Algebra Appl. 469, 364 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Guo, Y.: Constructing the unextendible maximally entangled basis from the maximally entangled basis. Phys. Rev. A 94, 052302 (2016)ADSCrossRefGoogle Scholar
  19. 19.
    Guo, Y., Wu, S.: Unextendible entangled bases with fixed Schmidt number. Phys. Rev. A 90, 054303 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    Guo, Y., Jia, Y., Li, X.: Multipartite unextendible entangled basis. Quantum Inf. Process. 14, 3553 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Yan-Ling Wang
    • 1
  • Mao-Sheng Li
    • 2
  • Shao-Ming Fei
    • 3
    • 4
  • Zhu-Jun Zheng
    • 1
  1. 1.Department of MathematicsSouth China University of TechnologyGuangzhouChina
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  3. 3.School of Mathematical SciencesCapital Normal UniversityBeijingChina
  4. 4.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations