Acta Phys. Hung. B 20/3–4 (2004) 261–268 QUANTUM ELECTRONICS Optical, Symplectic and Fresnel Tomographies of Quantum States ∗ Margarita A. Man’ko, 1 Sergio De Nicola, 2 Renato Fedele 3 and Vladimir I. Man’ko 1 1 P.N. Lebedev Physical Institute, Leninskii Prospect, 53, 119991 Moscow, Russia 2 Istituto di Cibernetica “Eduardo Caianiello” del CNR Comprensorio “A. Olivetti” Fabbr. 70, Via Campi Flegrei, 34 I-80078 Pozzuoli (NA), Italy 3 Dipartimento di Scienze Fisiche, Universit` a “Federico II” di Napoli and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte Sant Angelo, Via Cintia, I-80126 Napoli, Italy Received 11 October 2004 Abstract. The description of photon quantum states by means of probability- distribution functions (tomograms) of three different kinds (optical, symplectic and Fresnel ones) is presented. Mutual relations between the optical, sym- plectic and Fresnel tomograms are established. Evolution equation for states of Bose–Einstein condensate (Gross–Pitaevskii nonlinear equation) is given in the tomographic-probability representation. Entropy of solitons related to the Shannon entropy of the tomographic-probability representation is considered. Keywords: quantum tomography, symplectic tomography, Fresnel tomography, nonlinear Schr¨ odinger equation, solitons, Gross–Pitaevskii equation, Bose–Einstein condensates PACS: 03.65.Wj, 03.75.Lm * Based on the talk held at the 11th Central European Workshop on Quantum Optics (CEWQO11), 18–20 July 2004, Trieste, Italy. 1. Introduction There exist different invertable maps of functions, which are solutions of either linear or nonlinear differential equations, e.g. the linear map of the function onto its Fourier component. Recently a new type of linear map (called tomographic map) was suggested [1]. This map was elaborated for the wave function (or density matrix) of a quantum state. In this approach, the complex function (complex den- sity matrix) is mapped onto positive probability distribution of random coordinate 1589-9535/04/ $ 20.00 c  2004 Akad´ emiai Kiad´o, Budapest