Geometrical phases for the G(4,2) Grassmannian manifold

Journal of Mathematical Physics

Volumn 44, Issue 6, 2003, Pages 2463-2470

  Karle, Regina ^a   Pachos, Jiannis ^b,c  

^a UNIVERSITY OF MUNICH   (Germany)

^b MAX PLANCK INSTITUT FÜR QUANTENOPTIK   (Germany)

^c IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038345172     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1572551     Document Type: Article

References (21)

1. M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984).
2. F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984); for a review see, Geometric Phases in Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989).
3. F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984); for a review see, Geometric Phases in Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989).
4. P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999); S. Seshadri, S. Lakshmibala, and V. Balakrishnan, Phys. Rev. A 55, 869 (1997); K. Fujii, "Introduction to Grassmann manifolds and quantum computation," to appear in J. Appl. Math., quant-ph/0103011; E. Ercolessi, G. Marmo, G. Morandi, and N. Mukunda, "Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach," quant-ph/0105007.
5. P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999); S. Seshadri, S. Lakshmibala, and V. Balakrishnan, Phys. Rev. A 55, 869 (1997); K. Fujii, "Introduction to Grassmann manifolds and quantum computation," to appear in J. Appl. Math., quant-ph/0103011; E. Ercolessi, G. Marmo, G. Morandi, and N. Mukunda, "Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach," quant-ph/0105007.
6. P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999); S. Seshadri, S. Lakshmibala, and V. Balakrishnan, Phys. Rev. A 55, 869 (1997); K. Fujii, "Introduction to Grassmann manifolds and quantum computation," to appear in J. Appl. Math., quant-ph/0103011; E. Ercolessi, G. Marmo, G. Morandi, and N. Mukunda, "Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach," quant-ph/0105007.
7. P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999); S. Seshadri, S. Lakshmibala, and V. Balakrishnan, Phys. Rev. A 55, 869 (1997); K. Fujii, "Introduction to Grassmann manifolds and quantum computation," to appear in J. Appl. Math., quant-ph/0103011; E. Ercolessi, G. Marmo, G. Morandi, and N. Mukunda, "Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach," quant-ph/0105007.
8. M. Nakahara, Geometry, Topology and Physics (IOP Publishing, Bristol, 1990).
9. J. Pachos, P. Zanardi, and M. Rasetti, Phys. Rev. A 61, 010305(R) (2000).
10. J. Pachos and S. Chountasis, Phys. Rev. A 62, 052318 (2000).
11. L. M. Duan, J. I. Cirac, and P. Zoller, Science 292, 1695 (2001).
12. J. Pachos and H. Walther, "Quantum computation with trapped ions in an optical cavity," Phys. Rev. Lett. 89, 187903 (2002); A. Recati, T. Calarco, P. Zanardi, J. I. Cirac, and P. Zoller, "Holonomic quantum computation with neutral atoms," quant-ph/0204030.
13. J. Pachos and H. Walther, "Quantum computation with trapped ions in an optical cavity," Phys. Rev. Lett. 89, 187903 (2002); A. Recati, T. Calarco, P. Zanardi, J. I. Cirac, and P. Zoller, "Holonomic quantum computation with neutral atoms," quant-ph/0204030.
14. P. Solinas, P. Zanardi, N. Zanghi, and F. Rossi, "Nonadiabatic geometrical quantum gates in semiconductor quantum dots," quant-ph/0301089; "Holonomic quantum gates: A semiconductor-based implementation," quant-ph/0301090.
15. P. Solinas, P. Zanardi, N. Zanghi, and F. Rossi, "Nonadiabatic geometrical quantum gates in semiconductor quantum dots," quant-ph/0301089; "Holonomic quantum gates: A semiconductor-based implementation," quant-ph/0301090.
16. M. S. Gulley, A. G. White, and D. F. V. James, "A Raman approach to quantum logic in calcium-like ions," quant-ph/ 0112117 and references therein.
17. For a generalization of the Abelian Stokes theorem to the non-Abelian case see R. L. Karp, F. Mansouri, and J. S. Ruo, J. Math. Phys. 40, 6033 (1999).
18. I. Fuentes-Guridi, J. Pachos, S. Bose, V. Vedral, and S. Choi, Phys. Rev. A 66, 022102 (2002).
19. R. G. Unanyan, B. W. Shore, and K. Bergmann, Phys. Rev. A 59, 2910 (1999).
20. J. Pachos and P. Zanardi, Int. J. Mod. Phys. B 15, 1257 (2001).
21. J. Pachos, "Quantum computation by geometrical means," published in the AMS Contemporary Math Series volume entitled "Quantum computation and quantum information science," quant-ph/0003150.