Vladimir I. Man’ko

Probability representation and new entropic uncertainty relations for symplectic and optical tomograms

Entropic uncertainty relations for Shannon entropies associated with tomographic probability distributions of continuous quadratures are reviewed. New entropie uncertainty relations in the form of inequalities for integrals containing the tomograms of quantum states and deformation parameter are obtained.

Tomography of solitons

We develop the tomographic representation of wavefunctions which are solutions of the generalized nonlinear Schrodinger equation (NLSE) and show its connection with the Weyl–Wigner map. The generalized NLSE is presented in the form of a nonlinear Fokker–Planck-type equation for the standard probability distribution function (tomogram). In particular, this theory is applied to solitons, where tomograms for envelope bright solitons of a family of modified NLSEs are presented and numerically evaluated. Examples of symplectic tomography and Fresnel tomography of linear and nonlinear signals are discussed.

A tomographic approach to quantum nonlocality

We propose a tomographic approach to the study of quantum nonlocality in continuously variable quantum systems. On one hand we derive a Bell-like inequality for measured tomograms, and on the other we introduce pseudospin operators whose statistics can be inferred from the data characterizing the reconstructed state, thus giving the possibility of using standard Bell inequalities. Illuminating examples are also discussed.

Entanglement in probability representation of quantum states and tomographic criterion of separability

Entangled and separable states of a bipartite (multipartite) system are studied in the tomographic representation of quantum states. Properties of tomograms (joint probability distributions) corresponding to entangled states are discussed. The connection with star-product quantization is presented. U(N)-tomography and spin tomography as well as the relation of the tomograms to positive and completely positive maps are considered. The tomographic criterion of separability (necessary and sufficient condition) is formulated in terms of the equality of the specific function depending on unitary group parameters and positive map semigroup parameters to unity. Generalized Werner states are used as an example.

Inner composition law of pure-spin states

Superposition principle for spin degrees of freedom is described in terms of density operators only using a formulated composition law of pure-state density operators. Decoherence phenomenon and visibility of the interference pattern are discussed.

Nonstationary linear spin systems in the probability representation

The probability representation of angular momentum states and the connection of the representation with the formalism of the star-product quantization procedure are reviewed. The Schrodinger equation of a general time-dependent Hamiltonian, linear in angular momentum operators, and the evolution equation for the density operator in the probability representation are solved analytically by means of the formalism of linear time-dependent constants of motion. These analytical solutions define wavefunctions and tomograms of states (generic Dicke states), which contain the atomic coherent states as a particular case. The statistical properties of these new states are also evaluated. General forms of analytically solvable Hamiltonians are established in terms of the Euler angle parametrization of the three-dimensional rotations.

INTERFERENCE EFFECTS IN F-DEFORMED FIELDS

We show how the introduction of an algebaric field deformation affects the interference phenomena. We also give a physical interpretation of the developed theory.

RELATIVISTIC PROPERTIES OF MARGINAL' DISTRIBUTIONS

We study the properties of marginal distributions-projections of the phase space representation of a physical system-under relativistic transforms. We consider the Galilei case as well as the Lorentz transforms exploiting the relativistic oscillator model used for describing the mass spectrum of elementary particles.

Dynamic symmetries and entropic inequalities in the probability representation of quantum mechanics

The probability representation of quantum and classical statistical mechanics is discussed. Symplectic tomography, center‐of‐mass tomography, and spin tomography are studied. The connection of tomographic probabilities with dynamic symmetries like symplectic group is considered. Entropic uncertainty relations and inequalities for spin tomograms are reviewed.

Optical, symplectic and fresnel tomographies of quantum states

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, symplectic 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.