International Journal of Theoretical Physics

, Volume 55, Issue 6, pp 2824–2836 | Cite as

Geometry and Symmetric Coherent States of Three Qubits Systems

  • Xiao-Kan Guo


In this paper, we first generalize the previous results that relate 1- and 2-qubit geometries to complex and quaternionic Möbius transformations respectively, to the case of 3-qubit states under octonionic Möbius transformations. This completes the correspondence between the qubit geometries and the four normed division algebras. Thereby, new systems of symmetric coherent states with 2 and 3 qubits can be constructed by mapping the spin coherent states to their antipodal symmetric ponits on the generalized Bloch spheres via Möbius transformations in corresponding dimensions. Finally, potential applications of the normed division algebras in physics are discussed.


Qubit geometry Möbius transformation Division algebra Symmetric coherent state 



The author thanks Zhiqiang Huang and Jin Yao for helpful discussions. This work is supported in part by Shanghai Normal University under the Grant No. A-6001-15-001487.


  1. 1.
    Baez, J.C.: The octonions. Bull. (New Ser.) Amer. Math. Soc. 39, 145 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bott, R., Milnor, J.: On the parallelizability of the spheres. Bull. Amer. Math. Soc. 64, 87 (1958)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Steenrod, N.: The Topology of Fibre Bundles. Princeton University Press, Princeton (1951)MATHGoogle Scholar
  4. 4.
    Mosseri, R., Dandoloff, R.: Geometry of entangled states, Bloch spheres and Hopf fibration. J. Phys. A: Math. Gen. 34, 10243 (2001)MathSciNetCrossRefMATHADSGoogle Scholar
  5. 5.
    Bernevig, B.A., Chen, H.-D.: Geometry of the three-qubit state, entanglement and division algebra. J. Phys. A: Math. Gen. 36, 8325 (2003)MathSciNetCrossRefMATHADSGoogle Scholar
  6. 6.
    Lee, J., Kim, C.H., Lee, E.K., Kim, J., Lee, S.: Qubit geometry and conformal mapping. Quantum Inf. Process. 1, 129 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Najarbashi, G., Ahadpour, S., Fasihi, M.A., Tavakoli, Y.: Geometry of a two-qubit state and intertwining quaternionic conformal mapping under local unitary transformations. J. Phys. A: Math. Theor. 40, 6481 (2007)MathSciNetCrossRefMATHADSGoogle Scholar
  8. 8.
    Najarbashi, G., Seifi, B., Mirzaei, S.: Two and three-qubits geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection. arXiv:1501.06013v2
  9. 9.
    Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)MathSciNetCrossRefADSGoogle Scholar
  10. 10.
    Manogue, C.A., Dray, T.: Octonionic Möbius transformations. Mod. Phys. Lett. A 14, 1243 (1999)MathSciNetCrossRefADSGoogle Scholar
  11. 11.
    Schray, J., Manogue, C.A.: Octonionic representations of Clifford algebras and triality. Found. Phys. 26, 17 (1996)MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Pashaev, O.K., Gurkan, Z.N.: Energy localization in maximally entangled two- and three-qubit phase space. New J. Phys. 14, 063007 (2012)CrossRefADSGoogle Scholar
  13. 13.
    Perelomov, A.M.: Coherent states for arbitrary lie group. Commun. Math. Phys. 26, 222 (1972)MathSciNetCrossRefMATHADSGoogle Scholar
  14. 14.
    Zhang, Y., Zhang, K.: Bell transform, teleportation operator and teleportation-based quantum computation. arXiv:1401.7009v3
  15. 15.
    Zhang, Z.-X.: Möbius transformation and Poisson integral representation for monogenic functions. Acta Math. Sin. (Chin. Ser.) 56, 487 (2013)MATHGoogle Scholar
  16. 16.
    Borsten, L., Dahanayake, D., Duff, M.J., Ebrahim, H., Rubens, W.: Black holes, qubits and octonions. Phys. Rep. 471, 113 (2009)MathSciNetCrossRefMATHADSGoogle Scholar
  17. 17.
    Grossman, B., Kephart, T.W., Stasheff, J.D.: Solutions to Yang-Mills field equations in eight dimensions and the last Hopf map. Commun. Math. Phys. 96, 431 (1984)MathSciNetCrossRefMATHADSGoogle Scholar
  18. 18.
    Acharyya, N., Chandra, N., Vadya, S.: Quantum entropy for the fuzzy sphere and its monopoles, JHEP11(2014)078Google Scholar
  19. 19.
    Deutsch, D.: Quantum computational networks. Proc. R. Soc. Lond. A 425, 73 (1989)MathSciNetCrossRefMATHADSGoogle Scholar
  20. 20.
    DiVincenzo, D.P.: Two-bit gates are universal for quantum computation. Phys. Rev. A 51, 1015 (1995)CrossRefADSGoogle Scholar
  21. 21.
    Zhang, Y.: Integrable quantum computation. Quantum Inf. Process. 12, 631 (2013)MathSciNetCrossRefMATHADSGoogle Scholar
  22. 22.
    Vlasov, A.Yu.: Clifford algebras and universal sets of quantum gates. Phys. Rev. A 63, 054302 (2001)MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Center for AstrophysicsShanghai Normal UniversityShanghaiChina

Personalised recommendations