Abstract
We present a necessary and sufficient product criterion for bipartite quantum states based on the rank of realignment matrix of density matrix. Then, this approach is generalized to multipartite systems. We first introduce the concept of semiproduct in a similar manner to the semiseparable and prove that semiproduct is equivalent to fully product. Therefore, a quantum state is bipartite product with respect to all possible partitions implies fully product which is different from the case of separability. For pure states, it can easily be seen that several necessary and sufficient separability criteria for multipartite systems are derived as a special case of our results. Several specific examples illustrate that our criteria are convenient and operational.
Similar content being viewed by others
References
Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)
Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661 (1991)
Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W. K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)
Gao, T., Yan, F.L., Li, Y.C.: Optimal controlled teleportation. Europhys. Lett. 84, 50001 (2008)
Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69, 2881 (1992)
Gross, C., Zibold, T., Nicklas, E., Esteve, J., Oberthaler, M.K.: Nonlinear atom interferometer surpasses classical precision limit. Nature 454, 1165 (2010)
Gurvits, L.: Classical complexity and quantum entanglement. J. Comput. Syst. Sci. 69, 448 (2004)
Hassan, A.S.M., Joag, P.S.: Separability criterion for multipartite quantum states based on the Bloch representation of density matrices. Quantum Inf. Comput. 8, 773 (2008)
Gabriel, A., Hiesmayr, B.C., Huber, M.: Criterion for k-separability in mixed multipartite systems. Quantum Inf. Comput. 10, 829 (2010)
Gao, T., Hong, Y.: Detection of genuinely entangled and nonseparable n-partite quantum states. Phys. Rev. A 82, 062113 (2010)
Gao, T., Hong, Y., Lu, Y., Yan, F.L.: Efficient k-separability criteria for mixed multipartite quantum states. Europhys. Lett. 104, 20007 (2013)
Hong, Y., Luo, S., Song, H.: Detecting k-nonseparability via quantum Fisher information. Phys. Rev. A 91, 042313 (2015)
Liu, L., Gao, T., Yan, F.L.: Separability criteria via sets of mutually unbiased measurements. Sci. Rep. 5, 13138 (2015)
Hong, Y., Luo, S.: Detecting k-nonseparability via local uncertainty relations. Phys. Rev. A 93, 042310 (2016)
Gao, T., Hong, Y.: Separability criteria for several classes of n-partite quantum states. Eur. Phys. J. D 61, 765 (2011)
Ma, Z.H., Chen, Z.H., Chen, J.L., Spengler, C., Gabriel, A., Huber, M.: Measure of genuine multipartite entanglement with computable lower bounds. Phys. Rev. A 83, 062325 (2011)
Hong, Y., Gao, T., Yan, F.L.: Measure of multipartite entanglement with computable lower bounds. Phys. Rev. A 86, 062323 (2012)
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)
Walter, M., Gross, D., Eisert, J.: Multi-partite entanglement, arXiv:https://arxiv.org/abs/1612.02437
Gühne, O., Tóth, G.: Entanglement detection. Phys. Rep. 474, 1 (2009)
Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: discord and related measures. Rev. Mod. Phys. 84, 1655 (2012)
Adesso, G., Bromley, T.R., Cianciaruso, M.: Measures and applications of quantum correlations. J. Phys. A: Math. Theor. 49, 473001 (2016)
Bera, A., Das, T., Sadhukhan, D., Roy, S.S., Sen (De), A., Sen, U.: Quantum discord and its allies: a review. arXiv:https://arxiv.org/abs/1703.10542
Moonen, M.S., Golub, G.H., de Moor, B.L.R.: In: Linear Algebra for Large Scale and Real-Time Applications, pp 293–314. Kluwer Publications (1993)
Pitsianis, N.P.: The Kronecker product in approximation and fast transform generation. Ph. D. Thesis, Cornell University, New York (1997)
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant Nos: 11371005, 11475054, and Hebei Natural Science Foundation of China under Grant No: A2016205145.
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Qi, X., Gao, T. & Yan, F. Necessary and Sufficient Product Criteria for Quantum States via the Rank of Realignment Matrix of Density Matrix. Int J Theor Phys 56, 3642–3648 (2017). https://doi.org/10.1007/s10773-017-3529-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-017-3529-x