We define a set of 2n−1−1 entanglement monotones for n qubits and give a single measure of entanglement in terms of these. This measure is zero except on globally entangled (fully inseparable) states. This measure is compared to the Meyer–Wallach measure for two, three, and four qubits. We determine the four-qubit state, symmetric under exchange of qubit labels, which maximizes this measure. It is also shown how the elementary monotones may be computed as a function of observable quantities. We compute the magnitude of our measure for the ground state of the four-qubit superconducting experimental system investigated in [M. Grajcar et al., Phys. Rev. Lett. 96, 047006 (2006)], and thus confirm the presence of global entanglement in the ground state.
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References
Einstein A., Podolsky B., Rosen N. (1935) Phys. Rev. 47, 777
Bell J.S. (1966) Rev. Mod. Phys. 38, 447
Aspect A., Grangier P., Roger G. (1981) Phys. Rev. Lett. 47, 460
Vidal G. (2003) Phys. Rev. Lett. 91, 147902
Gisin N., Ribordy G.G., Tittel W., Zbinden H. (2002) Rev. Mod. Phys. 74, 145
The straightforward generalizations to time-dependent Hamiltonians, thermal mixtures [see around (11)], incoherent evolution, etc. will not be given separately.
Carteret H.A., Higuchi A., Sudbery A. (2002) J. Math. Phys. 41, 7932
Linden N., Popescu S., Sudbery A. (1999) Phys. Rev. Lett. 83, 243
Wootters W.K. (1998) Phys. Rev. Lett. 80, 2245
D. A. Meyer and N. R. Wallach, In (R. K. Brylinski and G. Chen) The Mathematics of Quantum Computation (CRC Press, Boca Raton, 2002), p. 77.
Vidal G. (2000) J. Mod. Opt. 47, 355
Meyer D.A., Wallach N.R. (2002) J. Math. Phys. 43, 4273
Bourennane M. et al. (2005) Phys. Rev. Lett. 92, 087902
Wallach N.R. (2005) Acta. Appl. Math. 86, 203
Dur W., Vidal G., Cirac J.I. (2000) Phys. Rev. A 62, 062314
Brennen G.K. (2003) Quant. Inf. Comput. 3, 616
D. M. Greenberger, M. Horne, and A. Zeilinger, In M. Kafatos (ed.), Bell’s Theorem, Quantum Theory and Conceptions of the Universe, Kluwer, Boston (1989).
Nielsen M.A., Chuang I.L. (2000) Quantum computation, and quantum information. Cambridge University Press, Cambridge
Scott A.J. (2004) Phys. Rev. A 69, 052330
Yu C.-S., Song H.-S. (2006) Phys. Rev. A 73, 022325
A. Osterloh and J. Siewert, Phys. Rev. A 72, 012337 (2005); A. Osterloh and J. Siewert, Int. J. Quant. Inf. 4, 531 (2006), quant-ph/0506073.
Grajcar M. et al. (2006) Phys. Rev. Lett. 96, 047006
Greenberg Ya.S. et al. (2002) Phys. Rev. B 66, 224511
Farhi E., Goldstone J., Gutmann S., Lapan J., Lundgren A., Preda D. (2001) Science 292, 472
Havel T.F. (2002) Quant. Inf. Processing 1, 511
Aschauer H., Calsamiglia J., Hein M., Briegel H.J. (2004) Quant. Inf. Comput. 4, 383
G. A. Paz-Silva and J. H. Reina, quant-ph/0603102.
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Love, P.J., van den Brink, A.M., Smirnov, A.Y. et al. A Characterization of Global Entanglement. Quantum Inf Process 6, 187–195 (2007). https://doi.org/10.1007/s11128-007-0052-7
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DOI: https://doi.org/10.1007/s11128-007-0052-7
Keywords
- Meyer–Wallach measure
- elementary monotones
- entanglement monotones
- global entanglement
- four-qubit state
- qubit labels