Geodesic
Unnormalized tomograms and quasidistributions of quantum states
Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 2, pp. 328-342
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider tomograms and quasidistributions, such as the Wigner functions, the Glauber–Sudarshan $P$-functions, and the Husimi $Q$-functions, that violate the standard normalization condition for probability distribution fucntions. We introduce special conditions for the Wigner function to determine the tomogram with the Radon transform and study three different examples of states like the de Broglie plane wave, the Moschinsky shutter problem, and the stationary state of a charged particle in a uniform constant electric field. We show that their tomograms and quasidistribution functions expressed in terms of the Dirac delta function, the Airy function, and Fresnel integrals violate the standard normalization condition and the density matrix of the state therefore cannot always be reconstructed. We propose a method that allows circumventing this problem using a special tomogram in the limit form.
Keywords: quantum tomography, normalization condition, plane wave, Moschinsky shutter, particle in an electric field.
Mots-clés : quasidistribution 
@article{TMF_2018_197_2_a10,
     author = {V. I. Man'ko and L. A. Markovich},
     title = {Unnormalized tomograms and quasidistributions of quantum states},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {328--342},
     year = {2018},
     volume = {197},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_197_2_a10/}
}
TY  - JOUR
AU  - V. I. Man'ko
AU  - L. A. Markovich
TI  - Unnormalized tomograms and quasidistributions of quantum states
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 328
EP  - 342
VL  - 197
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_197_2_a10/
LA  - ru
ID  - TMF_2018_197_2_a10
ER  - 
%0 Journal Article
%A V. I. Man'ko
%A L. A. Markovich
%T Unnormalized tomograms and quasidistributions of quantum states
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 328-342
%V 197
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2018_197_2_a10/
%G ru
%F TMF_2018_197_2_a10
V. I. Man'ko; L. A. Markovich. Unnormalized tomograms and quasidistributions of quantum states. Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 2, pp. 328-342. http://geodesic.mathdoc.fr/item/TMF_2018_197_2_a10/

[1] E. Wigner, “On the quantum correction for thermodynamic equilibrium”, Phys. Rev., 40:5 (1932), 749–759 | DOI

[2] R. J. Glauber, “Photon correlations”, Phys. Rev. Lett., 10:3 (1963), 84–86 | DOI | MR

[3] E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams”, Phys. Rev. Lett., 10:7 (1963), 277–279 | DOI | MR

[4] K. Husimi, “Some formal properties of the density matrix”, Proc. Phys. Math. Soc. Japan, 22:4 (1940), 264–314 | Zbl

[5] Y. Hagiwaraa, Y. Hatta, “Use of the Husimi distribution for nucleon tomography”, Nucl. Phys. A., 940 (2015), 158–166 | DOI

[6] H.-W. Lee, “Theory and application of the quantum phase-space distribution functions”, Phys. Rep., 259:3 (1995), 147–211 | DOI | MR

[7] T. Kunihiroa, B. Muller, A. Ohnishi, A. Schafer, “Towards a theory of entropy production in the little and big bang”, Progr. Theor. Phys., 121:3 (2009), 555–575, arXiv: 0809.4831 | DOI

[8] S. Mancini, V. I. Man'ko, P. Tombesi, “Symplectic tomography as classical approach to quantum systems”, Phys. Lett. A, 213:1–2 (1996), 1–6 | DOI | MR

[9] S. Mancini, V. I. Man'ko, P. Tombesi, “Classical-like description of quantum dynamics by means of symplectic tomography”, Found. Phys., 27:6 (1997), 801–824 | DOI | MR

[10] J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten”, Ber. Verh. Sächs. Akad. Wiss Leipzig, 69 (1917), 262–277 | Zbl

[11] V. I. Man'ko, L. A. Markovich, “Symplectic tomography of de Broglie wave”, J. Russ. Laser Res., 38:6 (2017), 507–515 | DOI

[12] G. G. Amosov, V. I. Man'ko, “Characteristic functions of states in star-product quantization”, J. Russ. Laser Res., 30:5 (2009), 435–442 | DOI

[13] M. Moshinsky, “Diffraction in time”, Phys. Rev., 88:3 (1952), 625–631 | DOI | MR

[14] V. I. Man'ko, M. Moshinsky, A. Sharma, “Diffraction in time in terms of Wigner distributions and tomographic probabilities”, Phys. Rev. A, 59:3 (1999), 1809–1816, arXiv: quant-ph/9902075 | DOI

[15] F. Bayen, M. Flato, M. Fronsdal, C. Lichnerowicz, D. Sternheimer, “Quantum mechanics as a deformation of classical mechanics”, Lett. Math. Phys., 1:6 (1977), 521–530 | DOI | MR

[16] O. V. Man'ko, V. I. Man'ko, G. Marmo, “Star-product of generalized Wigner–Weyl symbols on $SU(2)$ group, deformations, and tomographic probability distribution”, Phys. Scr., 62:6 (2000), 446–452 | DOI | MR

[17] O. V. Man'ko, V. I. Man'ko, G. Marmo, C. Stornaiolo, “Radon transform of the Wheeler–De Witt equation and tomography of quantum states of the universe”, Gen. Rel. Grav., 37:1 (2005), 99–114 | DOI | MR

[18] O. V. Man'ko, V. I. Man'ko, G. Marmo, P. Vitale, “Star products, duality, double Lie algebras”, Phys. Lett. A., 360:4–5 (2007), 522–532 | DOI | MR

[19] K. E. Cahill, R. J. Glauber, “Ordered expansions in boson amplitude operators”, Phys. Rev., 177:5 (1969), 1857–1881 | DOI

[20] K. E. Cahill, R. J. Glauber, “Density operators, quasiprobability distributions”, Phys. Rev., 177:5 (1969), 1882–1902 | DOI

[21] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Obobschennye funktsii. Vyp. 5. Integralnaya geometriya i svyazannye s nei voprosy teorii predstavlenii, Fizmatlit, M., 1962 | MR | Zbl

[22] M. A. Man'ko, V. I. Man'ko, R. V. Mendes, “A probabilistic operator symbol framework for quantum information”, J. Russ. Laser Res., 27:6 (2006), 507–532 | DOI

[23] A. D. Poularikas, The Handbook of Formulas and Tables for Signal Processing, CRC Press, Boca Raton, 1999

[24] L. Schwartz, Théorie des distributions, Hermann, Paris, 1950 | MR

[25] D. Serre, Is square of Delta function defined somewhere?, MathOverflow, \par https://mathoverflow.net/questions/48067

[26] J. F. Colombeau, Multiplication of Distributions. A Tool in Mathematics, Numerical Engineering, Theoretical Physics, Lecture Notes in Mathematics, 1532, Springer, Berlin, 1992 | DOI | MR

[27] J. F. Colombeau, A. Meril, “Generalized functions and multiplication of distributions on $\mathscr C^\infty$ manifolds”, J. Math. Anal. Appl., 186:2 (1994), 357–364 | DOI | MR

[28] H. M. Nussenzveig, “Moshinsky functions, resonances and tunneling”, Symmetries in Physics, Proceedings of the International Symposium Held in Honor of Professor Marcos Moshinsky (Cocoyoc, Morelos, México, June 3–7, 1991), eds. A. Frank, K. B. Wolf, Springer, Berlin, Heidelberg, 1992, 293–310 | DOI | MR

[29] V. I. Man'ko, E. V. Shchukin, “A charged particle in an electric field in the probability representation of quantum mechanics”, J. Russ. Las. Res., 22:6 (2001), 545–560 | DOI

[30] O. Vallée, M. Soares, Ch. de Izarra, “An integral representation for the product of Airy functions”, Z. Angew. Math. Phys., 48:1 (1997), 156–160 | DOI | MR

[31] D. Dominici, R. S. Maier (eds.), Special Functions and Orthogonal Polynomials (University of Arizona, Tucson, AZ, April 21–22, 2007), Contemporary Mathematics, 471, AMS, Providence, RI, 2007 | MR

[32] O. Vallée, M. Soares, Airy Functions and Applications to Physics, World Sci., Singapore, 2004 | DOI | MR