Quantum Information Processing

, Volume 14, Issue 8, pp 2861–2881 | Cite as

A multipartite entanglement measure based on coefficient matrices

  • Chao Zhao
  • Guo-wu Yang
  • William N. N. Hung
  • Xiao-yu Li


The quantification of quantum entanglement has been extensively studied in past years. However, many existing entanglement measures are difficult to calculate. And lots of them are introduced only for bipartite system or only for the systems constituted by qubits. In this paper, we propose an entanglement measure for multipartite system based on vector lengths and the angles between vectors of the coefficient matrices. Our entanglement measure is simple and feasible, with a remarkable geometric meaning. Furthermore, we prove that our entanglement measure satisfies the three necessary conditions which are required for any entanglement measure: (1) It vanishes if and only if the state is (fully) separable; (2) it remains invariant under local unitary transformations; and (3) it cannot increase under local operation and classical communication. Finally, we apply our entanglement measure on some computational examples. It demonstrates that our entanglement measure is capable of dealing with quantum pure states with arbitrary dimensions and parties. Meanwhile, because it only needs to compute the vector lengths and the angles between vectors of every bipartition coefficient matrix, our entanglement measure is easy to calculate.


Multipartite entanglement Entanglement measure Coefficient matrices 



This work was supported by the National Natural Science Foundation of China under Grant No. 61272175.


  1. 1.
    Bennett, C.H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J.A., Wootters, W.K.: Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 78, 2031 (1996)ADSCrossRefGoogle Scholar
  2. 2.
    Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    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, 1895–1899 (1993)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  6. 6.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)MathSciNetADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Vedral, V., Plenio, M.B., Rippin, M.A., Knight, P.L.: Quantifying entanglement. Phys. Rev. Lett. 78, 2275–2279 (1997)MathSciNetADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Gühne, O., Tóth, G.: Entanglement detection. Phys. Rep. 474, 1–75 (2009)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Vidal, G.: Entanglement monotones. J. Mod. Opt. 47, 355–376 (2000)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Horodecki, M.: Simplifying monotonicity conditions for entanglement measures. Open Syst. Inf. Dyn. 12, 231–237 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Demkowicz-Dobrzanski, R., Buchleitner, A., Kus, M., Mintert, M.: Evaluable multipartite entanglement measures: multipartite concurrences as entanglement monotones. Phys. Rev. A 74, 052303 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    Hill, S., Wootters, W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022–5025 (1997)ADSCrossRefGoogle Scholar
  13. 13.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)ADSCrossRefGoogle Scholar
  14. 14.
    Long, Y., Qiu, D., Long, D.: An entanglement measure based on two order minors. J. Phys. A Math. Theor. 42, 265301 (2009)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Huang, Y., Qiu, D.W.: Concurrence vectors of multipartite states based on coefficient matrices. Quantum Inf Process. 11, 235–254 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Coffman, V., Kundu, G., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)ADSCrossRefGoogle Scholar
  17. 17.
    Miyake, A.: Classification of multipartite entangled states by multidimensional determinants. Phys. Rev. A 67, 012108 (2003)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Lohmayer, R., Osterloh, A., Siewert, J., Uhlmann, A.: Entangled three-qubit states without concurrence and three-tangle. Phys. Rev. Lett. 97, 260502 (2006)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Wong, A., Christensen, N.: Potential multiparticle entanglement measure. Phys. Rev. A 63, 044301 (2001)ADSCrossRefGoogle Scholar
  20. 20.
    Osterloh, A., Siewert, J.: Constructing \(N\)-qubit entanglement monotones from antilinear operators. Phys. Rev. A 72, 012337 (2005)ADSCrossRefGoogle Scholar
  21. 21.
    Fei, S.M., Zhao, M.J., Chen, K., Wang, Z.X.: Experimental determination of entanglement for arbitrary pure states. Phys. Rev. A 80, 032320 (2009)ADSCrossRefGoogle Scholar
  22. 22.
    Wei, T.C., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003)ADSCrossRefGoogle Scholar
  23. 23.
    Hassan, A., Joag, P.S.: Geometric measure for entanglement in \(N\)-qudit pure states. Phys. Rev. A 80, 042302 (2009)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Chen, L., Xu, A., Zhu, H.: Computation of the geometric measure of entanglement for pure multiqubit states. Phys. Rev. A 82, 032301 (2010)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Li, X.R., Li, D.F.: Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states. Phys. Rev. A 86, 042332 (2012)ADSCrossRefGoogle Scholar
  26. 26.
    Li, X.R., Li, D.F.: Classification of general \(n\)-qubit states under stochastic local operations and classical communication in terms of the rank of coefficient matrix. Phys. Rev. Lett. 108, 180502 (2012)ADSCrossRefzbMATHGoogle Scholar
  27. 27.
    Vidal, G.: Entanglement of pure states for a single copy. Phys. Rev. Lett. 83, 1046–1049 (1999)ADSCrossRefGoogle Scholar
  28. 28.
    Jonathan, D., Plenio, M.B.: Minimal conditions for local pure-state entanglement manipulation. Phys. Rev. Lett. 83, 1455–1458 (1999)MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Alberti, P.M., Uhlmann, A.: Stochasticity and Partial Order. Reidel, Dordrecht (1982)zbMATHGoogle Scholar
  30. 30.
    Uhlmann, A.: Fidelity and concurrence of conjugated states. Phys. Rev. A 62, 032307 (2000)MathSciNetADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Rungta, P., Buzěk, V., Caves, C.M., Hillery, M., Milburn, G.J.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Audenaert, K., Verstraete, F., Moor, B.D.: Variational characterizations of separability and entanglement of formation. Phys. Rev. A 64, 052304 (2001)ADSCrossRefGoogle Scholar
  33. 33.
    Badziag, P., Deuar, P.: Concurrence in arbitrary dimensions. J. Mod. Opt. 49, 1289 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Albeverio, S., Fei, S.M.: A note on invariants and entanglements. J. Opt. B Quantum Semiclass. Opt. 3, 223–227 (2001)MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    Fan, H., Matsumoto, K., Imai, H.: Quantify entanglement by concurrence hierarchy. J. Phys. A Math. Gen. 36, 4151–4158 (2003)MathSciNetADSCrossRefzbMATHGoogle Scholar
  36. 36.
    Li, Y.Q., Zhu, G.Q.: Concurrence vectors for entanglement of high-dimensional systems. Front. Phys. China 3(3), 250–257 (2008)ADSCrossRefGoogle Scholar
  37. 37.
    Bhaktavatsala Rao, D.D., Ravishankar, V.: A redefinition of concurrence and its generalisation to bosonic subsystems of N qubit systems (2003). Preprint quant-ph/0309047
  38. 38.
    Akhtarshenas, S.J.: Concurrence vectors in arbitrary multipartite quantum systems. J. Phys. A Math. Gen. 38, 6777–6784 (2005)MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    Heydari, H.: Concurrence for general multipartite states. J. Phys. A Math. Gen. 39, 15225–15229 (2006)MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    Heydari, H.: Entanglement witnesses and concurrence for multi-qubit states. Quantum Inf. Comput. 8(89), 0791–0796 (2008)MathSciNetGoogle Scholar
  41. 41.
    Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)MathSciNetADSCrossRefzbMATHGoogle Scholar
  42. 42.
    Brown, I.D., Stepney, S., Sudbery, A., Braunstein, S.L.: Searching for highly entangled multi-qubit states. J. Phys. A Math. Gen. 38, 1119–1131 (2005)MathSciNetADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Li, H., Wang, S.H., Cui, J.L., Long, G.L.: Quantifying entanglement of arbitrary-dimensional multipartite pure states in terms of the singular values of coefficient matrices. Phys. Rev. A 87, 042335 (2013)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Chao Zhao
    • 1
  • Guo-wu Yang
    • 1
  • William N. N. Hung
    • 2
  • Xiao-yu Li
    • 3
  1. 1.Big Data Research CenterUniversity of Electronic Science and Technology of ChinaChengduChina
  2. 2.Synopsys Inc.Mountain ViewUSA
  3. 3.School of Information and Software EngineeringUniversity of Electronic Science and Technology of ChinaChengduChina

Personalised recommendations