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Quantum Information Processing

, Volume 11, Issue 2, pp 481–492 | Cite as

The n-tangle of odd n qubits

Article

Abstract

Coffman et al. presented the 3-tangle of three qubits in Phys Rev A 61, 052306 (2000). Wong and Christensen (Phys Rev A 63, 044301, 2001) extended the standard form of the 3-tangle to even number of qubits, known as n-tangle. In this paper, we propose a generalization of the standard form of the 3-tangle to any odd n-qubit pure states and call it the n-tangle of odd n qubits. We show that the n-tangle of odd n qubits is invariant under permutations of the qubits, and is an entanglement monotone. The n-tangle of odd n qubits can be considered as a natural entanglement measure of any odd n-qubit pure states, and used for stochastic local operations and classical communication classification.

Keywords

3-tangle n-tangle Concurrence Residual entanglement 

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References

  1. 1.
    Horodecki R., Horodecki P., Horodecki M., Horodecki K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Wootters W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)ADSCrossRefGoogle Scholar
  3. 3.
    Rungta P., Bužek V., Caves C.M., Hillery M., Milburn G.J.: Universal state of inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Uhlmann A.: Fidelity and concurrence of conjugated states. Phys. Rev. A 62, 032307 (2000)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Coffman V., Kundu J., Wootters W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)ADSCrossRefGoogle Scholar
  6. 6.
    Dür W., Vidal G., Cirac J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Lee S., Joo J., Kim J.: Entanglement of three-qubit pure states in terms of teleportation capability. Phys. Rev. A 72, 024302 (2005)ADSCrossRefGoogle Scholar
  8. 8.
    Emary C., Beenakker C.W.J.: Relation between entanglement measures and Bell inequalities for three qubits. Phys. Rev. A 69, 032317 (2004)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Ou Y., Fan H.: Monogamy inequality in terms of negativity for three-qubit states. Phys. Rev. A 75, 062308 (2007)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Miyake A.: Classification of multipartite entangled states by multidimensional determinants. Phys. Rev. A 67, 012108 (2003)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Wong A., Christensen N.: Potential multiparticle entanglement measure. Phys. Rev. A 63, 044301 (2001)ADSCrossRefGoogle Scholar
  12. 12.
    Sharma S.S., Sharma N.K.: Four-tangle for pure states. Phys. Rev. A 82, 012340 (2010)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Li D., Li X., Huang H., Li X.: Stochastic local operations and classical communication invariant and the residual entanglement for n qubits. Phys. Rev. A 76, 032304 (2007) [arXiv:quant-ph/0704.2087]ADSCrossRefGoogle Scholar
  14. 14.
    Li D., Li X., Huang H., Li X.: An entanglement measure for n qubits. J. Math. Phys. 50, 012104 (2009)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Yu C., Song H.: Multipartite entanglement measure. Phys. Rev. A 71, 042331 (2005)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Osborne T.J., Verstraete F.: General monogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett. 96, 220503 (2006)ADSCrossRefGoogle Scholar
  17. 17.
    Li X., Li D.: Relationship between the n-tangle and the residual entanglement of even n qubits. Quantum Inf. Comput. 10, 1018 (2010)MathSciNetADSMATHGoogle Scholar
  18. 18.
    Gour G., Wallach N.R.: All maximally entangled four-qubit states. J. Math. Phys. 51, 112201 (2010)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Luque J.-G., Thibon J.-Y., Toumazet F.: Unitary invariants of qubit systems. Math. Struct. in Comp. Science 17, 1133 (2007)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Li D., Li X., Huang H., Li X.: Stochastic local operations and classical communication properties of the n-qubit symmetric Dicke states. Europhys. Lett. 87, 20006 (2009)ADSCrossRefGoogle Scholar
  21. 21.
    Li D., Li X., Huang H., Li X.: Simple criteria for the SLOCC classification. Phys. Lett. A 359, 428 (2006)ADSMATHCrossRefGoogle Scholar
  22. 22.
    Li D., Li X., Huang H., Li X.: Classification of four-qubit states by means of a stochastic local operation and the classical communication invariant and semi-invariants. Phys. Rev. A 76, 052311 (2007)ADSCrossRefGoogle Scholar
  23. 23.
    Li D., Li X., Huang H., Li X.: SLOCC classification for nine families of four-qubits. Quantum Inf. Comput. 9, 0778 (2009)Google Scholar
  24. 24.
    Li X., Li D.: Stochastic local operations and classical communication equations and classification of even n qubits. J. Phys. A: Math. Theor. 44, 155304 (2011) [arXiv:quant-ph/0910.4276]ADSCrossRefGoogle Scholar
  25. 25.
    Buniy, R.V., Kephart, T.W.: An algebraic classification of entangled states. arXiv:quant-ph/1012.2630Google Scholar
  26. 26.
    Viehmann O., Eltschka C., Siewert J.: Polynomial invariants for discrimination and classification of four-qubit entanglement. Phys. Rev. A 83, 052330 (2011)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of mathematical sciencesTsinghua UniversityBeijingChina

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