Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 011, 11 pages      arXiv:1008.4836      https://doi.org/10.3842/SIGMA.2011.011

Entanglement of Grassmannian Coherent States for Multi-Partite n-Level Systems

Ghader Najarbashi and Yusef Maleki
Department of Physics, University of Mohaghegh Ardabili, Ardabil, 179, Iran

Received September 05, 2010, in final form January 19, 2011; Published online January 24, 2011

Abstract
In this paper, we investigate the entanglement of multi-partite Grassmannian coherent states (GCSs) described by Grassmann numbers for n>2 degree of nilpotency. Choosing an appropriate weight function, we show that it is possible to construct some well-known entangled pure states, consisting of GHZ, W, Bell, cluster type and bi-separable states, which are obtained by integrating over tensor product of GCSs. It is shown that for three level systems, the Grassmann creation and annihilation operators b and b together with bz form a closed deformed algebra, i.e., SUq(2) with q=ei/3, which is useful to construct entangled qutrit-states. The same argument holds for three level squeezed states. Moreover combining the Grassmann and bosonic coherent states we construct maximal entangled super coherent states.

Key words: entanglement; Grassmannian variables; coherent states.

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References

  1. Nielsen M.A., Chuang I.L., Quantum computation and quantum information, Cambridge University Press, Cambridge, 2000.
  2. Petz D., Quantum information theory and quantum statistics, Springer-Verlag, Berlin, 2008.
  3. van Enk S.J., Decoherence of multidimensional entangled coherent states, Phys. Rev. A 72 (2005), 022308, 6 pages, quant-ph/0503207.
  4. van Enk S.J., Hirota O., Entangled coherent states: teleportation and decoherence, Phys. Rev. A 64 (2001), 022313, 6 pages, quant-ph/0012086.
  5. Fujii K., Introduction to coherent states and quantum information theory, quant-ph/0112090.
  6. Najarbashi G., Maleki Y., Maximal entanglement of two-qubit states constructed by linearly independent coherent states, arXiv:1007.1387.
  7. Fu H., Wang X., Solomon A.I., Maximal entanglement of nonorthogonal states: classification, Phys. Lett. A 291 (2001), 73-76, quant-ph/0105099.
  8. Wang X., Sanders B.C., Multipartite entangled coherent states, Phys. Rev. A 65 (2001), 012303, 7 pages, quant-ph/0104011.
  9. Wang X., Bipartite entangled non-orthogonal states, J. Phys. A: Math. Gen. 35 (2002), 165-173, quant-ph/0102011.
  10. Wang X., Sanders B.C., Pan S.-H., Entangled coherent states for systems with SU(2) and SU(1,1) symmetries, J. Phys. A: Math. Gen. 33 (2000), 7451-7467, quant-ph/0001073.
  11. Wang X., Quantum teleportation of entangled coherent states, Phys. Rev. A 64 (2001), 022302, 4 pages, quant-ph/0102048.
  12. Majid S., Rodríguez-Plaza M.J., Random walk and the heat equation on superspace and anyspace, J. Math. Phys. 35 (1994), 3753-3760.
  13. Cabra D.C., Moreno E.F., Tanasa A., Para-Grassmann variables and coherent states, SIGMA 2 (2006), 087, 8 pages, hep-th/0609217.
  14. Najarbashi G., Fasihi M.A., Fakhri H., Generalized Grassmannian coherent states for pseudo-Hermitian n-level systems, J. Phys. A: Math. Theor. 43 (2010), 325301, 10 pages, arXiv:1007.1392.
  15. Borsten L., Dahanayake D., Duff M.J., Rubens W., Superqubits, Phys. Rev. D 81 (2010), 105023, 16 pages, arXiv:0908.0706.
  16. Khanna F.C., Malbouisson J.M.C., Santana A.E., Santos E.S., Maximum entanglement in squeezed boson and fermion states, Phys. Rev. A 76 (2007), 022109, 5 pages, arXiv:0709.0716.
  17. Castellani L., Grassi P A., Sommovigo L., Quantum computing with superqubits, arXiv:1001.3753.
  18. Najarbashi G., Fasihi M.A., Mirmasoudi F., Mirzaei S., Entanglement of fermionic coherent states for pseudo Hermitian Hamiltonian, Poster at International Iran Conference on Quantum Information-2010 (2010, Kish Island, Iran).
  19. Najarbashi G., Maleki Y., Entanglement in multi-qubit pure fermionic coherent states, arXiv:1004.3703.
  20. Cahill K.E., Glauber R.J., Density operators for fermions, Phys. Rev. A 59 (1999), 1538-1555, physics/9808029.
  21. Kerner R., Z3-graded algebras and the cubic root of the supersymmetry translations, J. Math. Phys. 33 (1992), 403-411.
  22. Filippov A.T., Isaev A.P., Kurdikov A.B., Para-Grassmann differential calculus, Theoret. and Math. Phys. 94 (1993), 150-165, hep-th/9210075.
  23. Isaev A.P., Para-Grassmann integral, discrete systems and quantum groups, Internat. J. Modern Phys. A 12 (1997), 201-206, q-alg/9609030.
  24. Cugliandolo L.F., Lozano G.S., Moreno E.F., Schaposnik F.A., A note on Gaussian integrals over para-Grassmann variables, Internat. J. Modern Phys. A 19 (2004), 1705-1714, hep-th/0209172.
  25. Ilinski K.N., Kalinin G.V., Stepanenko A.S., q-functional Wick's theorems for particles with exotic statistics, J. Phys. A: Math. Gen. 30 (1997), 5299-5310, hep-th/9704181.
  26. Barnum H., Knill E., Ortiz G., Somma R., Viola L., A subsystem-independent generalization of entanglement, Phys. Rev. Lett. 92 (2004), 107902, 4 pages, quant-ph/0305023.
  27. Munhoz P.P., Semião F.L., Vidiella-Barranco A., Cluster-type entangled coherent states, Phys. Lett. A 372 (2008), 3580-3585, arXiv:0705.1549.
  28. Fujii K., A relation between coherent states and generalized Bell states, quant-ph/0105077.
  29. Gerry C.C., Peart M., Spin squeezing and entanglement via hole-burning in atomic coherent states, Phys. Lett. A 372 (2008), 6480-6483.
  30. Sun C., Xue K., Wang G., Wu C., A study on the relations between the topological parameter and entanglement, arXiv:1001.4587.
  31. Ichikawa T., Sasaki T., Tsutsui I., Yonezawa N., Exchange symmetry and multipartite entanglement, Phys. Rev. A 78 (2008), 052105, 8 pages, arXiv:0805.3625.
  32. Mandilara A., Akulin V.M., Smilga A.V., Viola L., Quantum entanglement via nilpotent polynomials, Phys. Rev. A 74 (2006), 022331, 34 pages, quant-ph/0508234.


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