Generalized quantum tomographic maps

Some nonlinear generalizations of classical Radon tomography were introduced by Asorey et al (2008 Phys. Rev. A 77 042115), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas and circles) or quadratic surfaces (ellipsoids, hyperboloids and spheres). We consider here the quantum version of this novel nonlinear approach and obtain, by the systematic use of the Weyl map, a tomographic encoding approach to quantum states. Nonlinear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme and the reconstruction formulae of the density operators are explicitly given in a closed form, with an explicit construction of quantizers and dequantizers. The role of symmetry groups behind the generalized tomographic maps is analyzed in some detail. We also introduce new generalizations of the standard singular dequantizers of the symplectic tomographic schemes, where the Dirac delta-distributions of operator-valued arguments are replaced by smooth window functions, giving rise to the new concept of thick quantum tomography. Applications in quantum state measurements of photons and matter waves are discussed.