Abstract
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.
Similar content being viewed by others
References
A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991)
A.K. Ekert, J.G. Rarity, P.R. Tapster, G. Massimo Palma, Phys. Rev. Lett. 69, 1293 (1992)
W. Tittel, J. Brendel, H. Zbinden, N. Gisin, Phys. Rev. Lett. 84, 4737 (2000)
P.G. Kwiat, J. Mod. Opt. 44, 2173 (1997)
J.T. Barreiro, N.K. Langford, N.A. Peters, P.G. Kwiat, Phys. Rev. Lett. 95, 260501 (2005)
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)
R.W. Boyd, Nonlinear Optics (Academic Press, 2008)
J.P. Torres, A. Alexandrescu, Lluis Torner, Phys. Rev. A 68, 050301 (2003)
F.M. Miatto, A.M. Yao, S.M. Barnett, Phys. Rev. A 83, 033816 (2011)
A.M. Yao, New J. Phys. 13, 053048 (2011)
S.M. Barnett, Quantum Information (Oxford University Press, Oxford, 2009)
S.M. Barnett, S.J.D. Phoenix, Phys. Rev. A 40, 2404 (1989)
M.J.W. Hall, Phys. Rev. A 55, 100 (1997)
M.J.W. Hall, E. Andersson, T. Brougham, Phys. Rev. A 74, 062308 (2006)
C.K. Hong, L. Mandel, Phys. Rev. A 31, 2409 (1985)
C.W. Monken, P.H. Souto Ribeiro, S. Padua, Phys. Rev. A 57, 3123 (1998)
C.K. Law, J.H. Eberly, Phys. Rev. Lett. 92, 127903 (2004)
M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)
C.K. Law, I.A. Walmsley, J.H. Eberly, Phys. Rev. Lett. 84, 5304 (2000)
A. Ekert, P.L. Knight, Am. J. Phys. 63, 415 (1995)
E. Abramochkin, V. Volostnikov, Opt. Commun. 83, 123 (1991)
S.S. Straupe, D.P. Ivanov, A.A. Kalinkin, I.B. Bobrov, S.P. Kulik, Phys. Rev. A 83, 060302 (2011)
L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A. 45, 8185 (1992)
C.E. Shannon, W. Weaver, The Mathematical Theory of Communication (University of Illinois Press, Urbana, 1949)
E.T. Jaynes, Phys. Rev. 106, 620 (1957)
T.M. Cover, J.A. Thomas, Elements of Information Theory (John Wiley & Sons, 1991)
A. Réyni, Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability (1960), p. 547
S.T. Flammia, A. Hamma, T.L. Hughes, X.G. Wen, Phys. Rev. Lett. 103, 261601 (2009)
C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54, 3824 (1996)
C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53, 2046 (1996)
S.M. Barnett, S.J.D. Phoenix, Phys. Rev. A 44, 535 (1991)
M.V. Fedorov, M.A. Efremov, P.A. Volkov, E.V. Moreva, S.S. Straupe, S.P. Kulik, Phys. Rev. A 77, 032336 (2008)
Y.M. Mikhailova, P.A. Volkov, M.V. Fedorov, Phys. Rev. A 78, 062327 (2008)
T. Brougham, S.M. Barnett, Phys. Rev. A 85, 032322 (2012)
X. Ma, C.-H.F. Fung, H.-K. Lo, Phys. Rev. A 76, 012307 (2007)
H. Di Lorenzo Pires, C.H. Monken, M.P. van Exter, Phys. Rev. A 80, 022307 (2009)
G.N. Watson, J. London Math. Soc. 8, 189 (1933)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Miatto, F.M., Brougham, T. & Yao, A.M. Cartesian and polar Schmidt bases for down-converted photons. Eur. Phys. J. D 66, 183 (2012). https://doi.org/10.1140/epjd/e2012-30063-y
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjd/e2012-30063-y