Cartesian and polar Schmidt bases for down-converted photons

How high dimensional entanglement protects the shared information from non-ideal measurements
Regular Article Topical issue: High Dimensional Quantum Entanglement. Guest editors: Sonja Franke-Arnold, Alessandra Gatti and Nicolas Treps


We derive an analytical form of the Schmidt modes of spontaneous parametric down-conversion (SPDC) biphotons in both Cartesian and polar coordinates. We show that these correspond to Hermite-Gauss (HG) or Laguerre-Gauss (LG) modes only for a specific value of their width, and we show how such value depends on the experimental parameters. The Schmidt modes that we explicitly derive allow one to set up an optimised projection basis that maximises the mutual information gained from a joint measurement. The possibility of doing so with LG modes makes it possible to take advantage of the properties of orbital angular momentum eigenmodes. We derive a general entropic entanglement measure using the Rényi entropy as a function of the Schmidt number, K, and then retrieve the von Neumann entropy, S. Using the relation between S and K we show that, for highly entangled states, a non-ideal measurement basis does not degrade the number of shared bits by a large extent. More specifically, given a non-ideal measurement which corresponds to the loss of a fraction of the total number of modes, we can quantify the experimental parameters needed to generate an entangled SPDC state with a sufficiently high dimensionality to retain any given fraction of shared bits.


Topical issue: High Dimensional Quantum Entanglement. Guest editors: Sonja Franke-Arnold, Alessandra Gatti and Nicolas Treps 


  1. 1.
    A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991)MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    A.K. Ekert, J.G. Rarity, P.R. Tapster, G. Massimo Palma, Phys. Rev. Lett. 69, 1293 (1992) ADSCrossRefGoogle Scholar
  3. 3.
    W. Tittel, J. Brendel, H. Zbinden, N. Gisin, Phys. Rev. Lett. 84, 4737 (2000) ADSCrossRefGoogle Scholar
  4. 4.
    P.G. Kwiat, J. Mod. Opt. 44, 2173 (1997) MathSciNetADSMATHGoogle Scholar
  5. 5.
    J.T. Barreiro, N.K. Langford, N.A. Peters, P.G. Kwiat, Phys. Rev. Lett. 95, 260501 (2005) ADSCrossRefGoogle Scholar
  6. 6.
    B. Jack, A.M. Yao, J. Leach, J. Romero, S. Franke-Arnold, D.G. Ireland, S.M. Barnett, M.J. Padgett, Phys. Rev. A 81, 043844 (2010) ADSCrossRefGoogle Scholar
  7. 7.
    R.W. Boyd, Nonlinear Optics (Academic Press, 2008)Google Scholar
  8. 8.
    J.P. Torres, A. Alexandrescu, Lluis Torner, Phys. Rev. A 68, 050301 (2003) ADSCrossRefGoogle Scholar
  9. 9.
    F.M. Miatto, A.M. Yao, S.M. Barnett, Phys. Rev. A 83, 033816 (2011) ADSCrossRefGoogle Scholar
  10. 10.
    A.M. Yao, New J. Phys. 13, 053048 (2011) ADSCrossRefGoogle Scholar
  11. 11.
    S.M. Barnett, Quantum Information (Oxford University Press, Oxford, 2009)Google Scholar
  12. 12.
    S.M. Barnett, S.J.D. Phoenix, Phys. Rev. A 40, 2404 (1989) MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    M.J.W. Hall, Phys. Rev. A 55, 100 (1997)ADSMATHCrossRefGoogle Scholar
  14. 14.
    M.J.W. Hall, E. Andersson, T. Brougham, Phys. Rev. A 74, 062308 (2006) MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    C.K. Hong, L. Mandel, Phys. Rev. A 31, 2409 (1985) ADSCrossRefGoogle Scholar
  16. 16.
    C.W. Monken, P.H. Souto Ribeiro, S. Padua, Phys. Rev. A 57, 3123 (1998) ADSCrossRefGoogle Scholar
  17. 17.
    C.K. Law, J.H. Eberly, Phys. Rev. Lett. 92, 127903 (2004) ADSCrossRefGoogle Scholar
  18. 18.
    M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)Google Scholar
  19. 19.
    C.K. Law, I.A. Walmsley, J.H. Eberly, Phys. Rev. Lett. 84, 5304 (2000) ADSCrossRefGoogle Scholar
  20. 20.
    A. Ekert, P.L. Knight, Am. J. Phys. 63, 415 (1995)MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    E. Abramochkin, V. Volostnikov, Opt. Commun. 83, 123 (1991)ADSCrossRefGoogle Scholar
  22. 22.
    S.S. Straupe, D.P. Ivanov, A.A. Kalinkin, I.B. Bobrov, S.P. Kulik, Phys. Rev. A 83, 060302 (2011) ADSCrossRefGoogle Scholar
  23. 23.
    L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A. 45, 8185 (1992) ADSCrossRefGoogle Scholar
  24. 24.
    C.E. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Urbana, 1949) Google Scholar
  25. 25.
    E.T. Jaynes, Phys. Rev. 106, 620 (1957) MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    T.M. Cover, J.A. Thomas, Elements of Information Theory (John Wiley & Sons, 1991)Google Scholar
  27. 27.
    A. Réyni, Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability (1960), p. 547 Google Scholar
  28. 28.
    S.T. Flammia, A. Hamma, T.L. Hughes, X.G. Wen, Phys. Rev. Lett. 103, 261601 (2009) ADSCrossRefGoogle Scholar
  29. 29.
    C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54, 3824 (1996) MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53, 2046 (1996) ADSCrossRefGoogle Scholar
  31. 31.
    S.M. Barnett, S.J.D. Phoenix, Phys. Rev. A 44, 535 (1991)ADSCrossRefGoogle Scholar
  32. 32.
    M.V. Fedorov, M.A. Efremov, P.A. Volkov, E.V. Moreva, S.S. Straupe, S.P. Kulik, Phys. Rev. A 77, 032336 (2008) ADSCrossRefGoogle Scholar
  33. 33.
    Y.M. Mikhailova, P.A. Volkov, M.V. Fedorov, Phys. Rev. A 78, 062327 (2008) ADSCrossRefGoogle Scholar
  34. 34.
    T. Brougham, S.M. Barnett, Phys. Rev. A 85, 032322 (2012) ADSCrossRefGoogle Scholar
  35. 35.
    X. Ma, C.-H.F. Fung, H.-K. Lo, Phys. Rev. A 76, 012307 (2007) ADSCrossRefGoogle Scholar
  36. 36.
    H. Di Lorenzo Pires, C.H. Monken, M.P. van Exter, Phys. Rev. A 80, 022307 (2009) ADSCrossRefGoogle Scholar
  37. 37.
    G.N. Watson, J. London Math. Soc. 8, 189 (1933)CrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.SUPA and Department of PhysicsUniversity of StrathclydeScotlandUK

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