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


SIGMA 7 (2011), 011, 11 pages      arXiv:1008.4836      http://dx.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.

pdf (354 Kb)   tex (14 Kb)

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.

Previous article   Next article   Contents of Volume 7 (2011)