Precise detection of multipartite entanglement in fourqubit Greenberger–Horne–Zeilinger diagonal states

Abstract

We propose a method of constructing the separability criteria for multipartite quantum states on the basis of entanglement witnesses. The entanglement witnesses are obtained by finding the maximal expectation values of Hermitian operators and then optimizing over all possible Hermitian operators. We derive a set of tripartite separability criteria for the four-qubit Greenberger–Horne–Zeilinger (GHZ) diagonal states. The derived criterion set contains four criteria that are necessary and sufficient for the tripartite separability of the highly symmetric four-qubit GHZ diagonal states; the criteria completely account for the numerically obtained boundaries of the tripartite separable state set. One of the criteria is just the tripartite separability criterion of the four-qubit generalized Werner states.

Keywords

multipartite entanglement / matched entanglement witness / GHZ diagonal state

References

[1] O. Gühne and G. Tóth, Entanglement detection, Phys. Rep. 474(1–6), 1 (2009)

[2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81(2), 865 (2009)

[3] M. Horodecki, P. Horodecki, and R. Horodecki, Separablity of mixed stated states: Necessary and sufficient conditions, Phys. Lett. A 223(1–2), 1 (1996)

[4] M. Horodecki, P. Horodecki, and R. Horodecki, Separability of n-particle mixed states: Necessary and sufficient conditions in terms of linear maps, Phys. Lett. A 283(1–2), 1 (2001)

[5] S. Gerke, J. Sperling, W. Vogel, Y. Cai, J. Roslund, N. Treps, and C. Fabre, Multipartite entanglement of a two-separable state, Phys. Rev. Lett. 117(11), 110502 (2016)

[6] F. Shahandeh, M. Ringbauer, J. C. Loredo, and T. C. Ralph, Ultrafine entanglement witnessing, Phys. Rev. Lett. 118(11), 110502 (2017)

[7] F. Buscemi, All entangled quantum states are nonlocal, Phys. Rev. Lett. 108(20), 200401 (2012)

[8] X. Chen, X. Y. Hu, and D. L. Zhou, Entanglement witness game, Phys. Rev. A 95(5), 052326 (2017)

[9] F. Baccari, D. Cavalcanti, P. Wittek, and A. Acín, Efficient device-independent entanglement detection for multipartite systems, Phys. Rev. X 7(2), 021042 (2017)

[10] L. Pezzè, Y. Li, W. Li, and A. Smerzi, Witnessing entanglement without entanglement witness operators, Proc. Natl. Acad. Sci. USA 113(41), 11459 (2016)

[11] J. Sperling and W. Vogel, Multipartite entanglement witnesses, Phys. Rev. Lett. 111(11), 110503 (2013)

[12] M. Huber and J. I. de Vicente, Structure of multidimensional entanglement in multipartite systems, Phys. Rev. Lett. 110(3), 030501 (2013)

[13] A. S. Sorensen and K. Molmer, Entanglement and extreme spin squeezing, Phys. Rev. Lett. 86(20), 4431 (2001)

[14] A. Kay, Optimal detection of entanglement in Greenberger–Horne–Zeilinger states, Phys. Rev. A 83, 020203(R) (2011)

[15] X. Y. Chen and L. Jiang, Graph-state basis for Pauli channels, Phys. Rev. A 83(5), 052316 (2011)

[16] H. Kaufmann, T. Ruster, C. T. Schmiegelow, M. A. Luda, V. Kaushal, J. Schulz, D. von Lindenfels, F. Schmidt-Kaler, and U. G. Poschinger, Scalable creation of long-lived multipartite entanglement, Phys. Rev. Lett. 119(15), 150503 (2017)

[17] Z.-E. Su, W.-D. Tang, D. Wu, X.-D. Cai, T. Yang, L. Li, N.-L. Liu, C.-Y. Lu, M. Zukowski, and J.-W. Pan, Experimental test of the irreducible four-qubit Greenberger–Horne–Zeilinger paradox, Phys. Rev. A 95, 030103 (2017)

[18] O. Gühne, Entanglement criteria and full separability of multi-qubit quantum states, Phys. Lett. A 375(3), 406 (2011)

[19] X. Y. Chen, L. Z. Jiang, P. Yu, and M. Tian, Necessary and sufficient fully separable criterion and entanglement for three-qubit Greenberger–Horne–Zeilinger diagonal states, Quantum Inform. Process. 14(7), 2463 (2015)

[20] X. Y. Chen, L. Z. Jiang, and Z. A. Xu, Matched witness for multipartite entanglement, Quantum Inform. Process. 16(4), 95 (2017)

[21] O. Gühne and M. Seevinck, Separability criteria for genuine multipartite entanglement, New J. Phys. 12(5), 053002 (2010)

[22] A. O. Pittenger and M. H. Rubin, Note on Separability of the Werner states in arbitrary dimensions, Opt. Commun. 179(1–6), 447 (2000)

RIGHTS & PERMISSIONS

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature