Quantum Information Processing

, Volume 12, Issue 1, pp 269–278 | Cite as

Examining the dimensionality of genuine multipartite entanglement

  • Christoph Spengler
  • Marcus HuberEmail author
  • Andreas Gabriel
  • Beatrix C. Hiesmayr


Entanglement in high-dimensional many-body systems plays an increasingly vital role in the foundations and applications of quantum physics. In the present paper, we introduce a theoretical concept which allows to categorize multipartite states by the number of degrees of freedom being entangled. In this regard, we derive computable and experimentally friendly criteria for arbitrary multipartite qudit systems that enable to examine in how many degrees of freedom a mixed state is genuine multipartite entangled.


Entanglement measures, witnesses, and other characterizations Algebraic methods Quantum information Foundations of quantum mechanics Formalism 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sachdev S.: Quantum Phase Transitions. Cambridge University Press, Cambridge (1999)Google Scholar
  2. 2.
    Akoury D., Kreidi K., Jahnke T., Weber T., Staudte A.: The simplest double slit: interference and entanglement in double photoionization of H2. Science 318(5852), 949–952 (2007)ADSCrossRefGoogle Scholar
  3. 3.
    Hiesmayr B.C.: Nonlocality and entanglement in a strange system. Eur. Phys. J. C 50, 73–79 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    Sarovar M., Ishizaki A., Fleming G.R., Whaley K.B.: Quantum entanglement in photosynthetic light-harvesting complexes. Nat. Phys. 6, 462–467 (2010)CrossRefGoogle Scholar
  5. 5.
    Gühne O., Toth G.: Entanglement detection. Phys. Rep. 474(1–6), 1–75 (2009)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Mermin N.D.: Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65, 3373–3376 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Pan J.-W., Bouwmeester D., Daniell M., Weinfurter A., Zeilinger H.: Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement. Nature 403(6769), 515–519 (2000)ADSCrossRefGoogle Scholar
  8. 8.
    Huber M., Friis N., Gabriel A., Spengler C., Hiesmayr B.C.: Lorentz invariance of entanglement classes in multipartite systems. Europhys. Lett. 95, 20002-p1–20002-p5 (2011)CrossRefGoogle Scholar
  9. 9.
    Raussendorf R., Briegel H.-J.: A one-way quantum computer. Phys. Rev. Lett. 86, 5188–5191 (2001)ADSCrossRefGoogle Scholar
  10. 10.
    Hillery M., Bužek V., Berthiaume A.: Quantum secret sharing. Phys. Rev. A 59, 1829–1834 (1999)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Schauer S., Huber M., Hiesmayr B.C.: Experimentally feasible security check for n-qubit quantum secret sharing. Phys. Rev. A 82, 062311 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Liu Z., Fan H.: Decay of multiqudit entanglement. Phys. Rev. A 79, 064305 (2009)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Collins D., Gisin N., Linden N., Massar S., Popescu S.: Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett. 88, 040404 (2002)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Cerf N.J., Bourennane M., Karlsson A., Gisin N.: Security of quantum key distribution using d-level systems. Phys. Rev. Lett. 88, 127902 (2002)ADSCrossRefGoogle Scholar
  15. 15.
    Keet A., Fortescue B., Markham D., Sanders B.C.: Quantum secret sharing with qudit graph states. Phys. Rev. A 82, 062315 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    Fitzi M., Gisin N., Maurer U.: Quantum solution to the Byzantine agreement problem. Phys. Rev. Lett. 87, 217901 (2001)ADSCrossRefGoogle Scholar
  17. 17.
    Li Y., Zeng G.-H.: Four-party quantum broadcast scheme based on Aharonov state. Commun. Theor. Phys. 50, 371 (2008)MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Looi S.Y., Yu L., Gheorghiu V., Griffiths R.B.: Quantum-error-correcting codes using qudit graph states. Phys. Rev. A 78, 042303 (2008)ADSCrossRefGoogle Scholar
  19. 19.
    Zhou D.L., Zeng B., Xu Z., Sun C.P.: Quantum computation based on d-level cluster state. Phys. Rev. A 68, 062303 (2003)ADSCrossRefGoogle Scholar
  20. 20.
    Joo J., Knight P.L., O’Brien J.L., Rudolph T.: One-way quantum computation with four-dimensional photonic qudits. Phys. Rev. A 76, 052326 (2007)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Bancal J.D., Brunner N., Gisin N., Liang Y.-C.: Detecting genuine multipartite quantum nonlocality: a simple approach and generalization to arbitrary dimensions. Phys. Rev. Lett. 106, 020405 (2011)ADSCrossRefGoogle Scholar
  22. 22.
    Son W., Lee J., Kim M.S.: Generic Bell inequalities for multipartite arbitrary dimensional systems. Phys. Rev. Lett. 96, 060406 (2006)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Cerf N.J., Massar S., Pironio S.: Greenberger–Horne–zeilinger paradoxes for many qudits. Phys. Rev. Lett. 89, 080402 (2002)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Lee J., Lee S.-W., Kim M.S.: Greenberger–Horne–Zeilinger nonlocality in arbitrary even dimensions. Phys. Rev. A 73, 032316 (2006)MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    Barreiro J.T., Langford N.K., Peters N.A., Kwiat P.G.: Generation of hyperentangled photon pairs. Phys. Rev. Lett. 95, 260501 (2005)ADSCrossRefGoogle Scholar
  26. 26.
    Terhal B.M., Horodecki P.: Schmidt number for density matrices. Phys. Rev. A 61, 040301(R) (2000)MathSciNetADSGoogle Scholar
  27. 27.
    Sanpera A., Bruss D., Lewenstein M.: Schmidt-number witnesses and bound entanglement. Phys. Rev. A 63, 050301(R) (2001)ADSCrossRefGoogle Scholar
  28. 28.
    Bruss D.: Characterizing entanglement. J. Math. Phys. 43, 4237–4251 (2002). doi: 10.1063/1.1494474 MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Eisert J., Briegel H.-J.: Schmidt measure as a tool for quantifying multiparticle entanglement. Phys. Rev. A 64, 022306 (2001)ADSCrossRefGoogle Scholar
  30. 30.
    Lim J., Ryu J., Yoo S., Lee C., Bang J., Lee J.: Genuinely high-dimensional nonlocality optimized by complementary measurements. New J. Phys. 12, 103012 (2010)ADSCrossRefGoogle Scholar
  31. 31.
    Li C.-M., Chen K., Reingruber A., Chen Y.-N., Pan J.-W.: Verifying genuine high-order entanglement. Phys. Rev. Lett. 105, 210504 (2010)ADSCrossRefGoogle Scholar
  32. 32.
    Huber M., Mintert F., Gabriel A., Hiesmayr B.C.: Detection of high-dimensional genuine multipartite entanglement of mixed states. Phys. Rev. Lett. 104, 210501 (2010)ADSCrossRefGoogle Scholar
  33. 33.
    Huber M., Erker P., Schimpf H., Gabriel A., Hiesmayr B.C.: Experimentally feasible set of criteria detecting genuine multipartite entanglement in n-qubit Dicke states and in higher-dimensional systems. Phys. Rev. A 83, 040301(R) (2011)ADSGoogle Scholar
  34. 34.
    Huber M., Schimpf H., Gabriel A., Spengler C., Bruss D., Hiesmayr B.C.: Experimentally implementable criteria revealing substructures of genuine multipartite entanglement. Phys. Rev. A 83, 022328 (2011)ADSCrossRefGoogle Scholar
  35. 35.
    Gabriel A., Huber M., Radic S., Hiesmayr B.C.: Computable criterion for partial entanglement in continuous-variable quantum systems. Phys. Rev. A 83, 052318 (2011)ADSCrossRefGoogle Scholar
  36. 36.
    Jungnitsch B., Moroder T., Gühne O.: Taming multiparticle entanglement. Phys. Rev. Lett. 106, 190502 (2011)ADSCrossRefGoogle Scholar
  37. 37.
    Kim J.S., Sanders B.C.: Generalized W-class state and its monogamy relation. J. Phys. A Math. Theor. 41, 495301 (2008)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Spengler C., Huber M., Hiesmayr B.C.: A composite parameterization of unitary groups, density matrices and subspaces. J. Phys. A Math. Theor. 43, 385306 (2010)MathSciNetADSCrossRefGoogle Scholar
  39. 39.
    Spengler C., Huber M., Hiesmayr B.C.: A geometric comparison of entanglement and quantum nonlocality in discrete systems. J. Phys. A Math. Theor. 44, 065304 (2011)MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    Spengler C., Huber M., Hiesmayr B.C.: Composite parameterization and Haar measure for all unitary and special unitary groups. J. Math. Phys. 53, 013501 (2012). doi: 10.1063/1.3672064 MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    Thew R.T., Nemoto K., White A.G., Munro W.J.: Qudit quantum-state tomography. Phys. Rev. A 66, 012303 (2002)MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    Ma Z.-H., Chen Z.-H., Chen J.-L., Spengler C., Gabriel A., Huber M.: Measure of genuine multipartite entanglement with computable lower bounds. Phys. Rev. A 83, 062325 (2011)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Christoph Spengler
    • 1
  • Marcus Huber
    • 1
    Email author
  • Andreas Gabriel
    • 1
  • Beatrix C. Hiesmayr
    • 1
  1. 1.Faculty of PhysicsUniversity of ViennaViennaAustria

Personalised recommendations