Geodesic
Entanglement of Grassmannian Coherent States for Multi-Partite $n$-Level Systems
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 $b^\dagger$ together with $b_{z}$ form a closed deformed algebra, i.e., $SU_{q}(2)$ with $q=e^{\frac{2\pi i}3}$, 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.
Keywords: entanglement; Grassmannian variables; coherent states. 
@article{SIGMA_2011_7_a10,
     author = {Ghader Najarbashi and Yusef Maleki},
     title = {Entanglement of {Grassmannian} {Coherent} {States} for {Multi-Partite} $n${-Level} {Systems}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2011},
     volume = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a10/}
}
TY  - JOUR
AU  - Ghader Najarbashi
AU  - Yusef Maleki
TI  - Entanglement of Grassmannian Coherent States for Multi-Partite $n$-Level Systems
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2011
VL  - 7
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a10/
LA  - en
ID  - SIGMA_2011_7_a10
ER  - 
%0 Journal Article
%A Ghader Najarbashi
%A Yusef Maleki
%T Entanglement of Grassmannian Coherent States for Multi-Partite $n$-Level Systems
%J Symmetry, integrability and geometry: methods and applications
%D 2011
%V 7
%U http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a10/
%G en
%F SIGMA_2011_7_a10
Ghader Najarbashi; Yusef Maleki. Entanglement of Grassmannian Coherent States for Multi-Partite $n$-Level Systems. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a10/

[1] Nielsen M. A., Chuang I. L., Quantum computation and quantum information, Cambridge University Press, Cambridge, 2000 | MR

[2] Petz D., Quantum information theory and quantum statistics, Springer-Verlag, Berlin, 2008 | MR | Zbl

[3] van Enk S. J., “Decoherence of multidimensional entangled coherent states”, Phys. Rev. A, 72 (2005), 022308, 6 pp., arXiv: quant-ph/0503207 | DOI

[4] van Enk S. J., Hirota O., “Entangled coherent states: teleportation and decoherence”, Phys. Rev. A, 64 (2001), 022313, 6 pp., arXiv: quant-ph/0012086 | DOI

[5] Fujii K., Introduction to coherent states and quantum information theory, arXiv: 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, arXiv: quant-ph/0105099 | DOI | MR | Zbl

[8] Wang X., Sanders B. C., “Multipartite entangled coherent states”, Phys. Rev. A, 65 (2001), 012303, 7 pp., arXiv: quant-ph/0104011 | DOI | MR

[9] Wang X., “Bipartite entangled non-orthogonal states”, J. Phys. A: Math. Gen., 35 (2002), 165–173, arXiv: quant-ph/0102011 | DOI | MR | Zbl

[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, arXiv: quant-ph/0001073 | DOI | MR | Zbl

[11] Wang X., “Quantum teleportation of entangled coherent states”, Phys. Rev. A, 64 (2001), 022302, 4 pp., arXiv: quant-ph/0102048 | DOI | MR

[12] Majid S., Rodríguez-Plaza M. J., “Random walk and the heat equation on superspace and anyspace”, J. Math. Phys., 35 (1994), 3753–3760 | DOI | MR | Zbl

[13] Cabra D. C., Moreno E. F., Tanasa A., “Para-Grassmann variables and coherent states”, SIGMA, 2 (2006), 087, 8 pp., arXiv: hep-th/0609217 | DOI | MR | Zbl

[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 pp., arXiv: 1007.1392 | DOI | MR | Zbl

[15] Borsten L., Dahanayake D., Duff M. J., Rubens W., “Superqubits”, Phys. Rev. D, 81 (2010), 105023, 16 pp., arXiv: 0908.0706 | DOI | MR

[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 pp., arXiv: 0709.0716 | DOI | MR

[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, arXiv: physics/9808029 | DOI

[21] Kerner R., “$Z_3$-graded algebras and the cubic root of the supersymmetry translations”, J. Math. Phys., 33 (1992), 403–411 | DOI | MR

[22] Filippov A. T., Isaev A. P., Kurdikov A. B., “Para-Grassmann differential calculus”, Theoret. and Math. Phys., 94 (1993), 150–165, arXiv: hep-th/9210075 | DOI | MR | Zbl

[23] Isaev A. P., “Para-Grassmann integral, discrete systems and quantum groups”, Internat. J. Modern Phys. A, 12 (1997), 201–206, arXiv: q-alg/9609030 | DOI | MR | Zbl

[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, arXiv: hep-th/0209172 | DOI | MR | Zbl

[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, arXiv: hep-th/9704181 | DOI | MR | Zbl

[26] Barnum H., Knill E., Ortiz G., Somma R., Viola L., “A subsystem-independent generalization of entanglement”, Phys. Rev. Lett., 92 (2004), 107902, 4 pp., arXiv: quant-ph/0305023 | DOI | MR

[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 | DOI | MR | Zbl

[28] Fujii K., A relation between coherent states and generalized Bell states, arXiv: 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 | DOI

[30] Sun C., Xue K., Wang G., Wu C., A study on the relations between the topological parameter and entanglement, arXiv: 1001.4587 | Zbl

[31] Ichikawa T., Sasaki T., Tsutsui I., Yonezawa N., “Exchange symmetry and multipartite entanglement”, Phys. Rev. A, 78 (2008), 052105, 8 pp., arXiv: 0805.3625 | DOI

[32] Mandilara A., Akulin V. M., Smilga A. V., Viola L., “Quantum entanglement via nilpotent polynomials”, Phys. Rev. A, 74 (2006), 022331, 34 pp., arXiv: quant-ph/0508234 | DOI