International Journal of Theoretical Physics

, Volume 57, Issue 8, pp 2496–2503 | Cite as

A Proof for the Existence of Nonsquare Unextendible Maximally Entangled Bases

  • Feng Liu


The nonsquare unextendible maximally entangled basis (nUMEB) is a set of orthonormal maximally entangled states in \(C^{d}\otimes C^{d^{\prime }}~(d^{\prime }> d)\) which have no additional maximally entangled vectors orthogonal to all of them. We study nUMEBs in arbitrary bipartite spaces and present a constructive proof of the existence of nUMEBs in \(C^{d}\otimes C^{d^{\prime }}~(d^{\prime }\geq 2d)\). Furthermore, a bound condition for the existence of nUMEBs for \(C^{d}\otimes C^{d^{\prime }}~(d^{\prime }\geq 2d)\) is obtained.


Unextendible maximally entangled basis Orthonormal maximally entangled states Maximally entangled vector 



This work is supported by the Natural Science Foundation of China (Grant No. 61771294) and Shandong Provincial Natural Science Foundation, China (Grant No. ZR2015FQ006).


  1. 1.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bravyi, S., Smolin, J.A.: Unextendible maximally entangled bases. Phys. Rev. A 84, 042306 (2011)ADSCrossRefGoogle Scholar
  3. 3.
    Li, M.S., Wang, Y.L., Zheng, Z.J.: Unextendible maximally entangled bases and mutually unbiased bases in \(C^{d}\otimes C^{d^{,}}\). Phys. Rev. A 89, 062313 (2014)ADSCrossRefGoogle Scholar
  4. 4.
    Liu, F., Gao, F., Qin, S.J., Xie, S.C., Wen, Q.Y.: Multipartite entanglement indicators based on monogamy relations of n-qubit symmetric states. Sci. Rep. 6, 20302 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    Horodecki, R., Horodecki, P., Horodecki, M.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benhelm, J., Kirchmair, G., Roos, C.F., Blatt, R.: Towards fault-tolerant quantum computing with trapped ions. Nat. Phys. 4, 463 (2008)CrossRefGoogle Scholar
  7. 7.
    Bennett, C.H., Brassard, G., Crepeau, 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wang, T.Y., Wen, Q.Y., Gao, F., Lin, S., Zhu, F. C.: Cryptanalysis and improvement of multiparty quantum secret sharing schemes. Phys. Lett. A 373, 65 (2008)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Barreiro, J.T., Wei, T.C., Kwia, P.G.: Beating the channel capacity limit for linear photonic superdense coding. Nat. Phys. 4, 282 (2010)CrossRefGoogle Scholar
  11. 11.
    Bennett, C.H., DiVincenzo, D.P.: Quantum information and computation. Nature 404, 247 (2000)ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jakobi, M., Simon, C., Gisin, N., et al.: Practical private database queries based on a quantum-key-distribution protocol. Phys. Rev. A 83, 022301 (2011)ADSCrossRefGoogle Scholar
  14. 14.
    Gao, F., Liu, B., Huang, W., Wen, Q.Y.: Postprocessing of the oblivious key in quantum private query. IEEE. J. Sel. Top. Quant. 21, 6600111 (2015)Google Scholar
  15. 15.
    Wei, C.Y., Cai, X. Q, Liu, B., et al.: A generic construction of quantum-oblivious-key-transfer-based private query with ideal database security and zero failure. IEEE Trans. Comput. 67, 2 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chen, B., Fei, S.M.: Unextendible maximally entangled bases and mutually unbiased bases. Phys. Rev. A 88, 034301 (2013)ADSCrossRefGoogle Scholar
  17. 17.
    Li, Z.G., Zhao, M.J., Fei, S.M., Fan, H., Liu, W.M.: Mixed maximally entangled states. Quant. Inf. Comput. 12, 0063 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Landau, L.J., Streater, R.F.: On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras. Linear Algebr. Appl. 193, 107 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebr. Appl. 10, 285 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    DiVincenzo, D.P., Fuchs, C.A., Mabuchi, H., Smolin, J.A., Thapliyal, A.V., Uhlmann, A: . In: Proceedings of Quantum Computing and Quantum Communications: First NASA International Conference. Palm Springs, LNCS 1509, p 247. Springer, Heidelberg (1999)Google Scholar
  21. 21.
    Smolin, J.A., Verstraete, F., Winter, A.: Entanglement of assistance and multipartite state distillation. Phys. Rev. A 72, 052317 (2005)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematic and Information ScienceShandong Technology and Business UniversityYantaiChina

Personalised recommendations