Abstract
We present a practical entanglement classification scheme for pure state in form of \(2\times L\times M\times N\) under the stochastic local operation and classical communication (SLOCC), where every inequivalent class of the entangled quantum states may be sorted out according to its standard form and the corresponding transformation matrix. This provides a practical method for determining the interconverting matrix between two SLOCC equivalent entangled states, and classification examples for some \(2\times 4\times M\times N\) systems are also presented.
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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, 1895–1899 (1993)
Ekert, A.K.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–663 (1991)
Bennett, C.H., Wiesner, S.J.: Communication via one- and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992)
Mattle, K., Weinfurter, H., Kwiat, P.G., Zeilinger, A.: Dense coding in experimental quantum communication. Phys. Rev. Lett. 76, 4656–4659 (1996)
Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)
Bin, L., Li, J.-L., Li, X., Qiao, C.-F.: Local unitary classification of arbitrary dimensional multipartite pure states. Phys. Rev. Lett. 108, 050501 (2012)
Verstraete, F., Dehaene, J., De Moor, B., Verschelde, H.: Four qubits can be entangled in nine different ways. Phys. Rev. A 65, 052112 (2002)
Bastin, T., Krins, S., Mathonet, P., Godefroid, M., Lamata, L., Solano, E.: Operational families of entanglement classes for symmetric N-qubit states. Phys. Rev. Lett. 103, 070503 (2009)
Cornelio, M.F., de Toledo Piza, A.F.R.: Classification of tripartite entanglement with one qubit. Phys. Rev. A 73, 032314 (2006)
Cheng, S., Li, J., Qiao, C.-F.: Classification of the entangled states of \(2\times \text{ N }\times \text{ N }\). J. Phys. A Math. Theor. 43, 055303 (2010)
Li, J.-L., Qiao, C.-F.: Classification of the entangled states \(2\times \text{ M }\times \text{ N }\). Quantum Inf. Proc. 12, 251–268 (2013)
Li, X., Li, J., Liu, B., Qiao, C.-F.: The parametric symmetry and numbers of the entangled class of \(2\times \text{ M }\times \text{ N }\) system. Sci. China G 54, 1471–1475 (2011)
Li, J.-L., Li, S.-Y., Qiao, C.-F.: Classification of the entangled states \(\text{ L }\times \text{ N }\times \text{ N }\). Phys. Rev. A 85, 012301 (2012)
Lamata, L., León, J., Salgado, D., Solano, E.: Inductive classification of multipartite entanglement under stochastic local operations and classical communication. Phys. Rev. A 74, 052336 (2006)
Lamata, L., León, J., Salgado, D., Solano, E.: Inductive entanglement classification of four qubits under stochastic local operations and classical communication. Phys. Rev. A 75, 022318 (2007)
Li, X., Li, D.: Classification of general n-qubit states under stochastic local operations and classical communication in terms of the rank of coefficient matrix. Phys. Rev. Lett. 108, 180502 (2012)
Wang, S., Lu, Y., Gao, M., Cui, J., Li, J.: Classification of arbitrary-dimensional multipartite pure states under stochastic local operations and classical communication using the rank of coefficient matrix. J. Phys. A Math. Theor. 46, 105303 (2013)
Wang, S., Lu, Y., Long, G.-L.: Entanglement classification of \(2\times 2\times 2\times \text{ d }\) quantum systems via the ranks of the multiple coefficient matrices. Phys. Rev. A 87, 062305 (2013)
Van Loan, C.F.: The ubiquitous Kronecker product. J. Comput. Appl. Math. 123, 85–100 (2000)
Zhang, T.-G., Zhao, M.-J., Li, M., Fei, S.-M., Li-Jost, X.: Criterion of local unitary equivalence for multipartite states. Phys. Rev. A 88, 042304 (2013)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University, Cambridge (1991)
Chen, K., Wu, L.-A.: A matrix realignment method for recognizing entanglement. Quantum Inf. Comput. 3, 193–202 (2003)
Albeverio, S., Cattaneo, L., Fei, S.-M., Wang, X.-H.: Equivalence of tripartite quantum states under local unitary transformations. Int. J. Quantum Inform. 03, 603–609 (2005)
Gou, S.-C.: Quantum behavior of a two-level atom interacting with two modes of light in a cavity. Phys. Rev. A 40, 5116–5128 (1989)
El-Orany, F.A.A.: Relationship between the atomic inversion and Wigner function for multiphoton multimode Jaynes–Cummings model. J. Phys. A Math. Gen. 37, 6157–6171 (2004)
Dhar, H.S., Chatterjee, A., Ghosh, R.: Mapping generalized Jaynes–Cummings interaction into correlated finite-sized systems. J. Phys. B At. Mol. Opt. Phys. 47, 135501 (2014)
Jaynes, E.T., Cummings, F.W.: Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE 51, 89–109 (1963)
Blais, A., Huang, R.-S., Wallraff, A., Girvin, S.M., Schoelkopf, R.J.: Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004)
Hartmann, M.J., Brandao, F.G.S.L., Plenio, M.B.: Strongly interacting polaritons in coupled arrays of cavities. Nat. Phys. 2, 849–855 (2006)
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This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11121092, 11175249, 11375200 and 11205239.
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Sun, LL., Li, JL. & Qiao, CF. Classification of the entangled states of \(2 \times L \times M \times N\) . Quantum Inf Process 14, 229–245 (2015). https://doi.org/10.1007/s11128-014-0828-5
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DOI: https://doi.org/10.1007/s11128-014-0828-5