Quantum Information Processing

, Volume 11, Issue 3, pp 685–710 | Cite as

Majorana representation of symmetric multiqubit states



As early as 1932, Majorana had proposed that a pure permutation symmetric state of N spin-\({\frac{1}{2}}\) particles can be represented by N spinors, which correspond geometrically to N points on the Bloch sphere. Several decades after its conception, the Majorana representation has recently attracted a great deal of attention in connection with multiparticle entanglement. A novel use of this representation led to the classification of entanglement families of permutation symmetric qubits—based on the number of distinct spinors and their arrangement in constituting the multiqubit state. An elegant approach to explore how correlation information of the whole pure symmetric state gets imprinted in its parts is developed for specific entanglement classes of symmetric states. Moreover, an elegant and simplified method to evaluate geometric measure of entanglement in N-qubit states obeying exchange symmetry has been developed based on the distribution of the constituent Majorana spionors over the unit sphere. Multiparticle entanglement being a key resource in several quantum information processing tasks, its deeper understanding is essential. In this review, we present a detailed description of the Majorana representation of pure symmetric states and its applicability in investigating various aspects of multiparticle entanglement.


Majorana representation Permutation symmetry Entanglement classification Irreducibility of correlations Multiqubit states 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Sackett C.A. et al.: Experimental entanglement of four particles. Nature (London) 404, 256–259 (2000)ADSCrossRefGoogle Scholar
  2. Roos C.F., Riebe M., Häffner H., Hänsel W., Benhelm J., Lancaster G.P.T., Becher C., Schmidt-Kaler F., Blatt R.: Control and measurement of three-qubit entangled states. Science 304, 1478–1480 (2004)ADSCrossRefGoogle Scholar
  3. Leibfried D. et al.: Creation of a six-atom ‘Schrodinger Cat’ state. Nature (London) 438, 639–642 (2005)ADSCrossRefGoogle Scholar
  4. Sorensen A., Duan L.-M., Cirac J.I., Zoller P.: Many-particle entanglement with Bose-Einstein condensates. Nature (London) 409, 63–66 (2001)ADSCrossRefGoogle Scholar
  5. Usha Devi A.R., Prabhu R., Rajagopal A.K.: Characterizing multiparticle entanglement in symmetric N-qubit states via negativity of covariance matrices. Phys. Rev. Lett. 98, 060501-060504 (2007)Google Scholar
  6. Greenberger D.M., Horne M.A., Shimony A., Zeilinger A.: Bell’s theorem without inequalities. Am. J. Phys. 58, 1131–1143 (1990)MathSciNetADSCrossRefGoogle Scholar
  7. Dicke R.H.: Coherence in spontaneous radiation processes. Phys. Rev. 93, 99–110 (1954)ADSMATHCrossRefGoogle Scholar
  8. Majorana E.: Atomi orientati in campo magnetico variabile. Nuovo Cimento 9, 43–50 (1932)MATHCrossRefGoogle Scholar
  9. Bloch F., Rabi I.I.: Atoms in variable magnetic fields. Rev. Mod. Phys. 17, 237–244 (1945)ADSCrossRefGoogle Scholar
  10. Penrose R.: Shadows of the Mind. Oxford University Press, Oxford (1994)Google Scholar
  11. Mäkelä H., Messina A.: N-qubit states as points on the Bloch sphere. Physica Scripta T 140, 014054 (2010)ADSCrossRefGoogle Scholar
  12. Dennis M.R.: Canonical representation of spherical functions: Sylvester’s theorem, Maxwell’s multipoles and Majorana’s sphere. J. Phys. A. Math. Gen. 37, 9487–9500 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  13. Swain, J.: The MR of spins and the relation between SU(∞) and SDiff(S 2), hep-th/arxiv:0405004Google Scholar
  14. Dennis M.R.: Correlations between Maxwell’s multipoles for Gaussian random functions on the sphere. J. Phys. A. Math.Gen. 38, 1653–1658 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  15. Zimba J.: “Anticoherent” spin states via the Majorana representation. EJTP 3, 143–156 (2006)MATHGoogle Scholar
  16. Bastin T., Krins S., Mathonet P., Godefroid M., Lamata L., Solano E.: Operational families of entanglement classes for symmetric N-qubit states. Phys. Rev. Lett. 103, 070503 (2009)ADSCrossRefGoogle Scholar
  17. Bastin, T., Mathonet, P., Solano, E.: Operational entanglement families of symmetric mixed N-qubit states, quant-ph/arxiv:1011.1243Google Scholar
  18. Usha Devi, A.R., Sudha, Rajagopal, A.K.: Determining the whole pure symmetric N-qubit state from its parts, quant-ph/arXiv:1003.2450Google Scholar
  19. Usha Devi, A.R., Sudha, Rajagopal, A.K.: Interconvertibility and irreducibility of permutation symmetric three qubit pure states, quant-ph/arXiv:1002.2820Google Scholar
  20. Markham, D.: Entanglement and symmetry in permutation symmetric states, quant-ph/arxiv: 1001.0343Google Scholar
  21. Aulbach M., Markham D., Murao M.: The maximally entangled symmetric state in terms of the geometric measure. New J. Phys. 12, 073025 (2010)ADSCrossRefGoogle Scholar
  22. Martin J., Giraud O., Braun P.A., Braun D., Bastin T.: Multiqubit symmetric states with high geometric entanglement. Phys. Rev. A 81, 062347 (2010)ADSCrossRefGoogle Scholar
  23. Aulbach, M., Markham, D., Murao, M.: Geometric entanglement of symmetric states and the Majorana representation, quant-ph/arxiv: 1010.4777Google Scholar
  24. Shimony A.: Degree of entanglement. Ann. Phys. NY Acad. Sci. 755, 675 (1995)MathSciNetADSCrossRefGoogle Scholar
  25. Wei T.C., Ericsson M., Goldbart P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003)ADSCrossRefGoogle Scholar
  26. Rose M.E.: Elementary Theory of Angular Momentum. Wiley, New York (1957)MATHGoogle Scholar
  27. Dür W., Vidal G., Cirac J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)MathSciNetADSCrossRefGoogle Scholar
  28. Verstraete F., Dehaene J., De Moor B., Verschelde H.: Four qubits can be entangled in nine different ways. Phys. Rev. A 65, 052112 (2002)MathSciNetADSCrossRefGoogle Scholar
  29. Lamata L., León J., Salgado D., Solano E.: Inductive entanglement classification of four qubits under stochastic local operations and classical communication. Phys. Rev. A 75, 022318 (2007)ADSCrossRefGoogle Scholar
  30. Maser, A.A., Wiegner, R., Schilling, U., Thiel, C., von Zanthier, J.: A versatile source of polarization-entangled photons, quant-ph/arXiv:0911.5115Google Scholar
  31. Kiesel N., Wieczorek W., Krins S., Bastin T., WeinfurterH. Solano E.: Operational multipartite entanglement classes for symmetric photonic qubit states. Phys. Rev. A 81, 032316 (2010)ADSCrossRefGoogle Scholar
  32. Mathonet P., Krins S., Godefroid M., Lamata L., Solano E., Bastin T.: Entanglement equivalence of N-qubit symmetric states. Phys. Rev. A 1, 052315 (2010)MathSciNetADSCrossRefGoogle Scholar
  33. Nielsen M., Chuang I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  34. Linden N., Popescu S., Wootters W.K.: Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices. Phys. Rev. Lett. 89, 207901 (2002)ADSCrossRefGoogle Scholar
  35. Linden N., Wootters W.K.: The parts determine the whole in a generic pure quantum state. Phys. Rev. Lett. 89, 277906 (2002)ADSCrossRefGoogle Scholar
  36. Jones N.S., Linden N.: Parts of quantum states. Phys. Rev. A 71, 012324 (2005)MathSciNetADSCrossRefGoogle Scholar
  37. Walck S.N., Lyons D.W.: Only n-qubit Greenberger-Horne-Zeilinger states are undetermined by their reduced density matrices. Phys. Rev. Lett. 100, 050501 (2008)MathSciNetADSCrossRefGoogle Scholar
  38. Walck S.N., Lyons D.W.: Only n-qubit Greenberger-Horne-Zeilinger states contain n-partite information. Phys. Rev. A 79, 032326 (2009)ADSCrossRefGoogle Scholar
  39. Parashar P., Rana S.: N-qubit W states are determined by their bipartite marginals. Phys. Rev. A 80, 012319 (2009)MathSciNetADSCrossRefGoogle Scholar
  40. Parashar P., Rana S.: Reducible correlations in Dicke states. J. Phys. A. Math. Theor. 42, 462003 (2009)ADSCrossRefGoogle Scholar
  41. Coleman A.J.: Structure of fermion density matrices. Rev. Mod. Phys. 35, 668 (1963)ADSCrossRefGoogle Scholar
  42. Colmenero F., Pérezdel Valle C., Valdemoro C.: Approximating q-order reduced density matrices in terms of the lower order ones I general relations. Phys. Rev. A 47, 971–978 (1993)ADSCrossRefGoogle Scholar
  43. Nakatsuji H., Yasuda K.: Direct determination of the quantum-mechanical density matrix using the density equation. Phys. Rev. Lett. 76, 1039 (1996)ADSCrossRefGoogle Scholar
  44. Yasuda K., Nakatsuji H.: Direct determination of the quantum-mechanical density matrix using the density equation II. Phys. Rev. A 56, 2648–2657 (1997)ADSCrossRefGoogle Scholar
  45. Mazziotti D.A.: Contracted Schrödinger equation: determining quantum energies and two-particle density matrices without wave functions. Phys. Rev. A 57, 4219–4234 (1998)ADSCrossRefGoogle Scholar
  46. Mazziotti D.A.: Pursuit of N-representability for the contracted Schrödinger equation through density-matrix reconstruction. Phys. Rev. A 60, 3618–3626 (1999)ADSCrossRefGoogle Scholar
  47. Linden, N., Popescu, S., Schumacher, B., Westmoreland, M.: Reversibility of local transformations of multiparticle entanglement Preprint quant-ph/9912039Google Scholar
  48. Bennett C.H., Popescu S., Rohrlich C., Smolin J.A., Thapliyal A.V.: Exact and asymptotic measures of multipartite pure-state entanglement. Phys. Rev. A 63, 012307 (2000)ADSCrossRefGoogle Scholar
  49. Wootters W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)ADSCrossRefGoogle Scholar
  50. Hill S., Wootters W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78, 5022 (1997)ADSCrossRefGoogle Scholar
  51. Rajagopal A.K., Rendell R.: Robust and fragile entanglement of three qubits: relation to permutation symmetry. Phys. Rev. A 65, 032328 (2002)MathSciNetADSCrossRefGoogle Scholar
  52. Coffman V., Kundu J., Wootters W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)ADSCrossRefGoogle Scholar
  53. Plenio M., Virmani S.: An introduction to entanglement measures. Quantum Inf. Comput. 7, 1–51 (2007)MathSciNetMATHGoogle Scholar
  54. Horodecki R., Horodecki P., Horodecki M., Horodecki K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  55. Hayashi M., Markham D., Murao M., Owari M., Virmani S.: Entanglement of multiparty-stabilizer, symmetric and antisymmetric states. Phys. Rev. A 77, 012104 (2008)ADSCrossRefGoogle Scholar
  56. Hayashi M., Markham D., Murao M., Owari M., Virmani S.: The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes. J. Math. Phys. 50, 122104 (2009)MathSciNetADSCrossRefGoogle Scholar
  57. Wei T.-C., Severini S.: Matrix permanent and quantum entanglement of permutation invariant states. J. Math. Phys. 51, 092203 (2010)MathSciNetADSCrossRefGoogle Scholar
  58. Hübener R., Kleinmann M., Wei T.-C., Guillén C.G., Gühne O.: Geometric measure of entanglement for symmetric states. Phys. Rev. A 80, 032324 (2009)MathSciNetADSCrossRefGoogle Scholar
  59. Arecchi F.T., Courtens E., Gilmore G., Thomas H.: Atomic coherent states in quantum optics. Phys. Rev. A 6, 2211 (1972)ADSCrossRefGoogle Scholar
  60. Ruskai M.B.: N completeness, N representability, and Geminal expansions. Phys. Rev. A 5, 1336 (1972)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of PhysicsBangalore UniversityBangaloreIndia
  2. 2.Inspire Institute Inc.AlexandriaUSA
  3. 3.Department of PhysicsKuvempu UniversityShimogaIndia
  4. 4.DAMTP, Centre for Mathematical SciencesCambridgeUK

Personalised recommendations