Quantum Information Processing

, Volume 13, Issue 3, pp 815–823 | Cite as

Lower bound of concurrence for qubit systems

  • Xue-Na Zhu
  • Shao-Ming Fei


We study the concurrence of four-qubit quantum states and provide analytical lower bounds of concurrence in terms of the monogamy inequality of concurrence for qubit systems. It is shown that these lower bounds are able to improve the existing bounds and detect entanglement better. The approach is generalized to arbitrary qubit systems.


Monogamy inequality Lower bounds of concurrence Four-qubit mixed quantum Arbitrary qubit systems 



The work is supported by NSFC under number 11275131.


  1. 1.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  2. 2.
    Einstein, A., Podolsky, B., Rosen, N.: Quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)CrossRefADSMATHGoogle Scholar
  3. 3.
    Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature (London) 416, 608 (2002)CrossRefADSGoogle Scholar
  4. 4.
    Werner, R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277 (1989)CrossRefADSGoogle Scholar
  5. 5.
    Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008)MathSciNetCrossRefADSMATHGoogle Scholar
  6. 6.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)CrossRefADSGoogle Scholar
  7. 7.
    Uhlmann, A.: Fidelity and concurrence of conjugated states. Phys. Rev. A 62, 032307 (2000)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    Rungta, P., Bužek, V., Caves, C.M., Hillery, M., Milburn, G.J.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Mintert, F., Kuś, M., Buchleitner, A.: Concurrence of mixed bipartite quantum states in arbitrary dimensions. Phys. Rev. Lett. 92, 167902 (2004)CrossRefADSGoogle Scholar
  10. 10.
    Chen, K., Albeverio, S., Fei, S.M.: Concurrence of arbitrary dimensional bipartite quantum states. Phys. Rev. Lett. 95, 040504 (2005)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Breuer, H.P.: Separability criteria and bounds for entanglement measures. J. Phys. A: Math. Gen. 39, 11847 (2006)MathSciNetCrossRefADSMATHGoogle Scholar
  12. 12.
    Breuer, H.P.: Optimal entanglement criterion for mixed quantum states. Phys. Rev. Lett. 97, 080501 (2006)CrossRefADSGoogle Scholar
  13. 13.
    de Vicente, J.I.: Lower bounds on concurrence and separability conditions. Phys. Rev. A 75, 052320 (2007)MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Zhang, C.J., Zhang, Y.S., Zhang, S., Guo, G.C.: Optimal entanglement witnesses based on local orthogonal observables. Phys. Rev. A 76, 012334 (2007)CrossRefADSGoogle Scholar
  15. 15.
    Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996)CrossRefADSGoogle Scholar
  16. 16.
    Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)MathSciNetCrossRefADSGoogle Scholar
  17. 17.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)CrossRefADSGoogle Scholar
  18. 18.
    Vollbrecht, K.G.H., Werner, R.F.: Entanglement measures under symmetry. Phys. Rev. A 64, 062307 (2001)CrossRefADSGoogle Scholar
  19. 19.
    Rungta, P., Caves, C.M.: Concurrence-based entanglement measures for isotropic states. Phys. Rev. A 67, 012307 (2003)CrossRefADSGoogle Scholar
  20. 20.
    Chen, K., Albeverio, S., Fei, S.M.: Concurrence-based entanglement measure for Werner States. Rep. Math. Phys. 58, 325 (2006)MathSciNetCrossRefADSMATHGoogle Scholar
  21. 21.
    Fei, S.M., Wang, Z.X., Zhao, H.: A note on entanglement of formation and generalized concurrence. Phys. Lett. A 329, 414 (2004)MathSciNetCrossRefADSMATHGoogle Scholar
  22. 22.
    Gao, X.H., Fei, S.M., Wu, K.: Lower bounds of concurrence for tripartite quantum systems. Phys. Rev. A 74, 050303 (2006)CrossRefADSGoogle Scholar
  23. 23.
    Gao, X.H., Fei, S.M.: Estimation of concurrence for multipartite mixed states. Eur. Phys. J. Special Topics 159, 71 (2008)CrossRefADSGoogle Scholar
  24. 24.
    Gao, X.H., Sergio, A., Chen, K., Fei, S.M., Li-Jost, X.Q.: Entanglement of formation and concurrence for mixed states. Front Comput. Sci. China 2(2), 114–128 (2008)CrossRefGoogle Scholar
  25. 25.
    Chen, Z.H., Ma, Z.H., Chen, J.L., Severini, S.: Improved lower bounds on genuine-multipartite-entanglement concurrence. Phys. Rev. A 85, 062320 (2012)CrossRefADSGoogle Scholar
  26. 26.
    Carvalho, A.R.R., Mintert, F., Buchleitner, A.: Decoherence and multipartite entanglement. Phys. Rev. Lett. 93, 230501 (2004)CrossRefADSGoogle Scholar
  27. 27.
    Aolita, L., Mintert, F.: Measuring multipartite concurrence with a single factorizable observable. Phys. Rev. Lett. 97, 050501 (2006)CrossRefADSGoogle Scholar
  28. 28.
    Osborne, T.J., Verstraete, F.: General monogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett. 96, 220503 (2006)CrossRefADSGoogle Scholar
  29. 29.
    Zhang, C.J., Gong, Y.X., Zhang, Y.S., Guo, G.C.: Observable estimation of entanglement for arbitrary finite-dimensional mixed states. Rev. A 78, 042308 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsSouth China University of TechnologyGuangzhouP.R.China
  2. 2.School of Mathematical SciencesCapital Normal UniversityBeijingChina
  3. 3.Max-Planck-Institute for Mathematics in the SciencesLeipzigGermany

Personalised recommendations