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On monogamy of four-qubit entanglement

  • S. Shelly Sharma
  • N. K. Sharma
Article
  • 47 Downloads

Abstract

Our main result is a monogamy inequality satisfied by the entanglement of a focus qubit (one-tangle) in a four-qubit pure state and entanglement of subsystems. Analytical relations between three-tangles of three-qubit marginal states, two-tangles of two-qubit marginal states and unitary invariants of four-qubit pure state are used to obtain the inequality. The contribution of three-tangle to one-tangle is found to be half of that suggested by a simple extension of entanglement monogamy relation for three qubits. On the other hand, an additional contribution due to a two-qubit invariant which is a function of three-way correlations is found. We also show that four-qubit monogamy inequality conjecture of Regula et al. (Phys Rev Lett 113:110501, 2014), in which three-tangles are raised to the power \(\frac{3}{2}\), does not estimate the residual correlations, correctly, for certain subsets of four-qubit states. A lower bound on residual four-qubit correlations is obtained.

Keywords

Monogamy inequality Quantum entanglement Determinants of negativity fonts Unitary invariants Four-qubit correlations 

Notes

Acknowledgements

Financial support from Universidade Estadual de Londrina PR, Brazil, is acknowledged.

References

  1. 1.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th edn. Cambridge University Press, New York (2011)zbMATHGoogle Scholar
  2. 2.
    Calabrese, P., Cardy, J., Tonni, E.: Entanglement negativity in quantum field theory. Phys. Rev. Lett. 109, 130502 (2012)ADSCrossRefGoogle Scholar
  3. 3.
    Sahling, S., Remenyi, G., Paulsen, C., Monceau, P., Saligrama, V., Marin, C., Revcolevschi, A., Regnault, L.P., Raymond, S., Lorenzo, J.E.: Experimental realization of long-distance entanglement between spins in antiferromagnetic quantum spin chains. Nat. Phys. 11, 255–260 (2015)CrossRefGoogle Scholar
  4. 4.
    Lambert, N., Chen, Y.N., Chen, Y.C., Li, C.M., Chen, G.Y., Nori, F.: Quantum biology. Nat. Phys. 9, 10–18 (2013)CrossRefGoogle Scholar
  5. 5.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)ADSCrossRefGoogle Scholar
  7. 7.
    Regula, B., Di Martino, S., Lee, S., Adesso, G.: Strong monogamy conjecture for multiqubit entanglement: the four-qubit case. Phys. Rev. Lett. 113, 110501 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    Regula, B., Di Martino, S., Lee, S., Adesso, G.: Erratum: Strong monogamy conjecture for multiqubit entanglement: the four-qubit case [Phys. Rev. Lett. 113, 110501 (2014)]. Phys. Rev. Lett. 116, 049902(E) (2016)Google Scholar
  9. 9.
    Regula, B., Osterloh, A., Adesso, G.: Strong monogamy inequalities for four qubits. Phys. Rev. A 93, 052338 (2016)ADSCrossRefGoogle Scholar
  10. 10.
    Karmakar, S., Sen, A., Bhar, A., Sarkar, D.: Strong monogamy conjecture in a four-qubit system. Phys. Rev. A 93, 012327 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    Sharma, S.S., Sharma, N.K.: Sequential generation of polynomial invariants and N-body non-local correlations. Quantum Inf. Process. 15, 4973 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sharma, S.S., Sharma, N.K.: Upper bound on three-tangles of reduced states of four-qubit pure states. Phys. Rev. A 95, 062311 (2017)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Sharma, S.S., Sharma, N.K.: Genuine four tangle for four qubit states. AIP Conf. Proc. 1633, 35 (2014)ADSGoogle Scholar
  14. 14.
    Sharma, S.S., Sharma, N.K.: Local unitary invariants for N-qubit pure states. Phys. Rev. A 82, 052340 (2010)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Luque, J.G., Thibon, J.Y.: Polynomial invariants of four qubits. Phys. Rev. A 67, 042303 (2003)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    He, H., Vidal, G.: Disentangling theorem and monogamy for entanglement negativity. Phys. Rev. A 91, 012339 (2015)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Hill, S., Wootters, W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022 (1997)ADSCrossRefGoogle Scholar
  19. 19.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Acin, A., Andrianov, A., Jane, E., Tarrach, R.: Three-qubit pure-state canonical forms. J. Phys. A Math. Gen. 34, 6725 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Verstraete, F., Dehaene, J., De Moor, B., Verschelde, H.: Four qubits can be entangled in nine different ways. Phys. Rev. A 65, 052112 (2002)ADSMathSciNetCrossRefGoogle Scholar

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

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Estadual de LondrinaLondrinaBrazil
  2. 2.Departamento de MatematicaUniversidade Estadual de LondrinaLondrinaBrazil

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