Abstract
We study the concurrence of four-qubit quantum states and provide analytical lower bounds of concurrence in terms of the monogamy inequality of concurrence for qubit systems. It is shown that these lower bounds are able to improve the existing bounds and detect entanglement better. The approach is generalized to arbitrary qubit systems.
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Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Einstein, A., Podolsky, B., Rosen, N.: Quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777 (1935)
Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature (London) 416, 608 (2002)
Werner, R.F.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277 (1989)
Amico, L., Fazio, R., Osterloh, A., Vedral, V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008)
Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)
Uhlmann, A.: Fidelity and concurrence of conjugated states. Phys. Rev. A 62, 032307 (2000)
Rungta, P., Bužek, V., Caves, C.M., Hillery, M., Milburn, G.J.: Universal state inversion and concurrence in arbitrary dimensions. Phys. Rev. A 64, 042315 (2001)
Mintert, F., Kuś, M., Buchleitner, A.: Concurrence of mixed bipartite quantum states in arbitrary dimensions. Phys. Rev. Lett. 92, 167902 (2004)
Chen, K., Albeverio, S., Fei, S.M.: Concurrence of arbitrary dimensional bipartite quantum states. Phys. Rev. Lett. 95, 040504 (2005)
Breuer, H.P.: Separability criteria and bounds for entanglement measures. J. Phys. A: Math. Gen. 39, 11847 (2006)
Breuer, H.P.: Optimal entanglement criterion for mixed quantum states. Phys. Rev. Lett. 97, 080501 (2006)
de Vicente, J.I.: Lower bounds on concurrence and separability conditions. Phys. Rev. A 75, 052320 (2007)
Zhang, C.J., Zhang, Y.S., Zhang, S., Guo, G.C.: Optimal entanglement witnesses based on local orthogonal observables. Phys. Rev. A 76, 012334 (2007)
Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996)
Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)
Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)
Vollbrecht, K.G.H., Werner, R.F.: Entanglement measures under symmetry. Phys. Rev. A 64, 062307 (2001)
Rungta, P., Caves, C.M.: Concurrence-based entanglement measures for isotropic states. Phys. Rev. A 67, 012307 (2003)
Chen, K., Albeverio, S., Fei, S.M.: Concurrence-based entanglement measure for Werner States. Rep. Math. Phys. 58, 325 (2006)
Fei, S.M., Wang, Z.X., Zhao, H.: A note on entanglement of formation and generalized concurrence. Phys. Lett. A 329, 414 (2004)
Gao, X.H., Fei, S.M., Wu, K.: Lower bounds of concurrence for tripartite quantum systems. Phys. Rev. A 74, 050303 (2006)
Gao, X.H., Fei, S.M.: Estimation of concurrence for multipartite mixed states. Eur. Phys. J. Special Topics 159, 71 (2008)
Gao, X.H., Sergio, A., Chen, K., Fei, S.M., Li-Jost, X.Q.: Entanglement of formation and concurrence for mixed states. Front Comput. Sci. China 2(2), 114–128 (2008)
Chen, Z.H., Ma, Z.H., Chen, J.L., Severini, S.: Improved lower bounds on genuine-multipartite-entanglement concurrence. Phys. Rev. A 85, 062320 (2012)
Carvalho, A.R.R., Mintert, F., Buchleitner, A.: Decoherence and multipartite entanglement. Phys. Rev. Lett. 93, 230501 (2004)
Aolita, L., Mintert, F.: Measuring multipartite concurrence with a single factorizable observable. Phys. Rev. Lett. 97, 050501 (2006)
Osborne, T.J., Verstraete, F.: General monogamy inequality for bipartite qubit entanglement. Phys. Rev. Lett. 96, 220503 (2006)
Zhang, C.J., Gong, Y.X., Zhang, Y.S., Guo, G.C.: Observable estimation of entanglement for arbitrary finite-dimensional mixed states. Rev. A 78, 042308 (2008)
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The work is supported by NSFC under number 11275131.
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Zhu, XN., Fei, SM. Lower bound of concurrence for qubit systems. Quantum Inf Process 13, 815–823 (2014). https://doi.org/10.1007/s11128-013-0693-7
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DOI: https://doi.org/10.1007/s11128-013-0693-7