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Relations among k-ME concurrence, negativity, polynomial invariants, and tangle

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Abstract

The k-ME concurrence as a measure of multipartite entanglement (ME) unambiguously detects all k-nonseparable states in arbitrary dimensions and satisfies many important properties of an entanglement measure. Negativity is a simple computable bipartite entanglement measure. Invariant and tangle are useful tools to study the properties of the quantum states. In this paper, we mainly investigate the internal relations among the k-ME concurrence, negativity, polynomial invariants, and tangle. Strong links between k-ME concurrence and negativity as well as between k-ME concurrence and polynomial invariants are derived. We obtain the quantitative relation between k-ME \((k=n)\) concurrence and negativity for all n-qubit states, give an exact value of the n-ME concurrence for the mixture of n-qubit GHZ states and white noise, and derive an connection between k-ME concurrence and tangle for n-qubit W state. Moreover, we find that for any 3-qubit pure state the k-ME concurrence \((k=2, 3)\) is related to negativity, tangle, and polynomial invariants, while for 4-qubit states the relations between k-ME concurrence (for \(k = 2, 4)\) and negativity, and between k-ME concurrence and polynomial invariants also exist. Our work provides clear quantitative connections between k-ME concurrence and negativity, and between k-ME concurrence and polynomial invariants.

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References

  1. Horodecki, R., et al.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  2. Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  3. Bennett, C.H., et al.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  4. Gao, T., Yan, F.L., Li, Y.C.: Optimal controlled teleportation. Europhys. Lett. 84, 50001 (2008)

    Article  ADS  Google Scholar 

  5. Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69, 2881 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  6. Gross, C., et al.: Nonlinear atom interferometer surpasses classical precision limit. Nature 464, 1165 (2010)

    Article  ADS  Google Scholar 

  7. Eltschka, C., Siewert, J.: Quantifying entanglement resources. J. Phys. A Math. Theor. 47, 424005 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  8. Bennett, C.H., et al.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  9. Hill, S., Wootters, W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022 (1997)

    Article  ADS  Google Scholar 

  10. Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)

    Article  ADS  Google Scholar 

  11. Rungta, P., et al.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  12. Carvalho, A.R.R., Mintert, F., Buchleitner, A.: Decoherence and multipartite entanglement. Phys. Rev. Lett. 93, 230501 (2004)

    Article  ADS  Google Scholar 

  13. Hong, Y., Gao, T., Yan, F.L.: Measure of multipartite entanglement with computable lower bounds. Phys. Rev. A 86, 062323 (2012)

    Article  ADS  Google Scholar 

  14. Gao, T., Hong, Y.: Detection of genuinely entangled and nonseparable \(n\)-partite quantum states. Phys. Rev. A 82, 062113 (2010)

    Article  ADS  Google Scholar 

  15. Gao, T., et al.: Efficient \(k\)-separability criteria for mixed multipartite quantum states. Europhys. Lett. 104, 20007 (2013)

    Article  ADS  Google Scholar 

  16. Gao, T., Yan, F.L., van Enk, S.J.: Permutationally invariant part of a density matrix and nonseparability of \(N\)-qubit states. Phys. Rev. Lett. 112, 180501 (2014)

    Article  ADS  Google Scholar 

  17. Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)

    Article  ADS  Google Scholar 

  18. Dür, W., et al.: Distillability and partial transposition in bipartite systems. Phys. Rev. A 61, 062313 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  19. Horodecki, M., Horodecki, P., Horodecki, R.: Mixed-state entanglement and distillation: is there a “bound” entanglement in nature? Phys. Rev. Lett. 80, 5239 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  20. Lee, S., et al.: Convex-roof extended negativity as an entanglement measure for bipartite quantum systems. Phys. Rev. A 68, 062304 (2003)

    Article  ADS  Google Scholar 

  21. Vidal, G.: Entanglement monotones. J. Mod. Opt. 47, 355 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  22. Sharma, S.S., Sharma, N.K.: Quantum coherences, \(K\)-way negativities and multipartite entanglement. Phys. Rev. A 77, 042117 (2008)

    Article  ADS  Google Scholar 

  23. Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  24. Kraus, B.: Local unitary equivalence and entanglement of multipartite pure states. Phys. Rev. A 82, 032121 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  25. Liu, B., et al.: Local unitary classification of arbitrary dimensional multipartite pure states. Phys. Rev. Lett. 108, 050501 (2012)

    Article  ADS  Google Scholar 

  26. Wang, S.H., Lu, Y., Long, G.L.: Entanglement classification of \(2\times 2 \times 2 \times d\) quantum systems via the ranks of the multiple coefficient matrices. Phys. Rev. A 87, 062305 (2013)

    Article  ADS  Google Scholar 

  27. Li, X.R., Li, D.F.: Polynomial invariants of degree \(4\) for even-\(n\) qubits and their applications in entanglement classification. Phys. Rev. A 88, 022306 (2013)

    Article  ADS  Google Scholar 

  28. Sanz, M., et al.: Entanglement classification with algebraic geometry. J. Phys. A Math. Theor. 50, 195303 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  29. Grassl, M., Rötteler, M., Beth, T.: Computing local invariants of quantum-bit systems. Phys. Rev. A 58, 1833 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  30. Linden, N., et al.: On multi-particle entanglement. Fortsch. Phys. 46, 567 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  31. Meyer, D.A., Wallach, N.R.: Global entanglement in multiparticle systems. J. Math. Phys. 43, 4273 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  32. Choi, J.H., Kim, J.S.: Negativity and strong monogamy of multiparty quantum entanglement beyond qubits. Phys. Rev. A 92, 042307 (2015)

    Article  ADS  Google Scholar 

  33. Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)

    Article  ADS  Google Scholar 

  34. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  35. Dür, W., Cirac, J.I.: Classification of multiqubit mixed states: separability and distillability properties. Phys. Rev. A 61, 042314 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  36. Gao, T., Hong, Y.: Separability criteria for several classes of \(n\)-partite quantum states. Eur. Phys. J. D 61, 765 (2011)

    Article  ADS  Google Scholar 

  37. Sudbery, A.: On local invariants of pure three-qubit states. J. Phys. A Math. Gen. 34, 643 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  38. Barnum, H., Linden, N.: Monotones and invariants for multi-particle quantum states. J. Phys. A Math. Gen. 34, 6787 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  39. Verstraete, F., et al.: Four qubits can be entangled in nine different ways. Phys. Rev. A 65, 052112 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  40. Gorbachev, V.N., et al.: Can the states of the W-class be suitable for teleportation? Phys. Lett. A 314, 267 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  41. Joo, J., et al.: Quantum teleportation via a W state. New J. Phys. 5, 136 (2003)

    Article  ADS  Google Scholar 

  42. Joo, J., et al.: Quantum secure communication with W states. arXiv:quant-ph/0204003

  43. Cabello, A.: Bell’s theorem with and without inequalities for the three-qubit Greenberger–Horne–Zeilinger and W states. Phys. Rev. A 65, 032108 (2002)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 11475054 and the Hebei Natural Science Foundation under Grant Nos. A2016205145 and A2018205125.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, China

    Limei Zhang & Ting Gao

  2. College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang, 050024, China

    Fengli Yan

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  1. Limei Zhang
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  2. Ting Gao
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  3. Fengli Yan
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Correspondence to Ting Gao.

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Zhang, L., Gao, T. & Yan, F. Relations among k-ME concurrence, negativity, polynomial invariants, and tangle. Quantum Inf Process 18, 194 (2019). https://doi.org/10.1007/s11128-019-2223-8

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