Abstract
By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix, and multiple rows and columns from vectorization of reduced density matrices, the authors of Shen (Phys. Rev. A 92: 042332, 2015) presented a family of separable criteria to improve the computable cross-norm or realignment criterion Rudolph (Phys. Rev. A 67: 032312, 2003); Chen (Quantum Inf. Comput. 3: 193-202, 2003). In this paper, we first show that these criteria achieve their optimization when the parameterized matrix is chosen to be a constant matrix. It is then proved that the optimized criterion is equivalent to the corresponding criterion with one additional row and one additional column. This reduces the computation cost, since the combined realignment matrix possesses a lower dimension. Finally, the optimized criterion is further used to achieve the separable criterion for multipartite quantum states, which, by using a numerical example, is more efficient than the corresponding previous criteria based on linear contraction methods and sequential realignment methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability statement
All data generated or analyzed during this study are included in this published article.
References
Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, Cambridge (2010)
Gurvits, L.: in Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing (ACM Press, New York, 2003), pp. 10-19
Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)
Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1–8 (1996)
Rudolph, O.: Some properties of the computable cross-norm criterion for separability. Phys. Rev. A 67, 032312 (2003)
Rudolph, O.: Further results on the cross norm criterion for separability. Quantum Inf. Process. 4, 219–239 (2005)
Chen, K., Wu, L.A.: A matrix realignment method for recognizing entanglement. Quantum Inf. Comput. 3, 193–202 (2003)
de Vicente, J.I.: Separability criteria based on the Bloch representation of density matrices. Quantum Inf. Comput. 7, 624 (2007)
de Vicente, J.I.: Further results on entanglement detection and quantification from the correlation matrix criterion. J. Phys. A: Math. Theor. 41, 065309 (2008)
Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)
Gühne, O., Tóth, G.: Entanglement detection. Phys. Rep. 474, 1–75 (2009)
Horodecki, P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333 (1997)
Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed quantum states: linear contractions and permutation. Open Syst. Inf. Dyn. 13, 103–111 (2006)
Chen, K., Wu, L.A.: The generalized partial transposition criterion for separability of multipartite quantum states. Phys. Lett. A 306, 14–20 (2002)
Zhang, Y.H., Lu, Y.Y., Wang, G.B., Shen, S.Q.: Realignment criteria for recognizing multipartite entanglement of quantum states. Quantum Inf. Process. 16, 106 (2017)
Shen, S.Q., Wang, M.Y., Li, M., Fei, S.M.: Separability criteria based on the realignment of density matrices and reduced density matrices. Phys. Rev. A 92, 042332 (2015)
Zhang, C.J., Zhang, Y.S., Zhang, S., Guo, G.C.: Entanglement detection beyond the computable cross-norm or realignment criterion. Phys. Rev. A 77, 060301(R) (2008)
Bennett, C.H., DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Phys. Rev. Lett. 82, 5385 (1999)
Werner, R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277 (1989)
Acknowledgements
The authors thank the referees and the editor for their invaluable comments. This work is supported by NSFC (11775306, 12075159), the Shandong Provincial Natural Science Foundation for Quantum Science (ZR2021LLZ002), and the Fundamental Research Funds for the Central Universities (19CX02050A).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shen, SQ., Chen, L., Hu, AW. et al. Optimization of realignment criteria and its applications for multipartite quantum states. Quantum Inf Process 21, 135 (2022). https://doi.org/10.1007/s11128-022-03463-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-022-03463-3