Abstract.
In this study the intimate connection is established between the Banach space wavelet reconstruction method on homogeneous spaces with both singular and nonsingular vacuum vectors, and some of the well known quantum tomographies, such as: Moyal-representation for a spin, discrete phase space tomography, tomography of a free particle, Homodyne tomography, phase space tomography and SU(1,1) tomography. And both the atomic decomposition and the Banach frame nature of these quantum tomographic examples are also revealed in details. Finally the connection between the wavelet formalism on Banach space and Q-function is discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S.G. Mallat, A Wavelet Tour of Signal Processing (Academic Press, New York, 1998)
R.K. Young, Wavelet Theory and its Applications (Kluwer Academic Publishers, 1993)
Y. Meyer, Wavelets: Algorithms and Applications (SIAM, Philadelphia, 1993)
P. Goupillaud, A. Grossmann, J. Morlet. Geoxploration 23, 85 (1984)
A. Grossmann, J. Morlet, T. Paul, J. Math. Phys. 26, 2473 (1985)
A. Grossmann, J. Morlet, T. Paul, Ann. Inst. H. Poincare 45, 293 (1986)
I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, 1992)
S.T. Ali, J-P Antoine, J-P.Gazeau, Coherent States, Wavelets and their Generalizations Springer (2000)
P. Gavruta, J. Math. Anal. Appl. 321, 469 (2006)
O. Christensen, C. Heil, Math. Nachr. 185, 33 (1997)
H.G. Feichtinger, K.H. Grochenig, J. Functional Anal 86, 307 (1989)
M.A. Man'ko, V.I. Man'ko, R.V. Mendes, J. Physics A: Math. and Gen. 34, 8321 (2001) 8321
V.V. Kisil, Acta Appl. Math. 59, 79 (1999)
E.B. Davis, Quantum Theory of Open System (Academic Press, 1976)
J. Várilly, J.M. Gracia-Bondía, Ann. Phys. 190, 107 (1989)
S. Heiss, S. Weigert, Phys. Rev. A 63, 012105 (2001)
C. Miquel, J.P. Paz, M. Saraceno, Phy. Rev. A 65, 062309 (2002)
G.M. D'Ariano, S. Mancini, V.I. Manko, P. Tombesi. J. Opt. B: Quantum and Semiclassical Opt. 8, 1017 (1996)
G.M. D'Ariano, M.G.A. Paris, M.F. Sacchi, Advances in Imaging and Electron Physics 128, 205 (2003)
S. Mancini, O.V. Manko, V.I. Manko, P. Tombesi, J. Phys. A: Math. Gen. 34, (16) 3461 (2001)
G.M. D'Ariano, E. De Vito, L. Maccone, Phys. Rev A 64, 033805 (2001)
U. Leonhardt, Measuring the Quantum State of Light (Cambridge University Press, Cambridge, UK, 1997)
A.A. Kirillov, Elements of the Theory of Representations (Springer-Verlag, Berlin, 1976)
S. Dahlke, G. Steidl, G. Teschke, Advances in Computational Mathematics 21, 147 (2004); S. Dahlke, G. Steidl, G. Teschke, Weighted Coorbit spaces and Banach Frames on Homogeneous Spaces, to appear in Journal of Fourier Analysis and Applications
G.W. Wei, Y.B. Zhao, Y. Xiang, I Theory and algorithm Int. J. Numer. Math. Eng. 55, 913 (2002)
R.J. Glauber, Phys. Rev. 130, 2529 (1963)
K. Husimi, Proc. Phys. Mat. Soc. Jpn 22, 264 (1940)
C.L. Mehta, J. Math. Phys. 5, 69 (1940)
J.G. Kirkwood, Phys. Rev. 44, 31 (1933)
W. Miller, Topics in harmonic analysis with applications to radar and sonar, Lecture note (2002) www.ima.umn.edu/ miller/radarla.pdf
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mirzaee, M., Rezaei, M. & Jafarizadeh, M. Quantum tomography with wavelet transform in Banach space on homogeneous space. Eur. Phys. J. B 60, 193–201 (2007). https://doi.org/10.1140/epjb/e2007-00330-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2007-00330-1