Generalized quantum tomographic maps

TL;DR

This paper extends quantum tomography by introducing non-linear, quadratic curve-based maps, providing explicit reconstruction formulas, and exploring symmetry groups, with applications to photon and matter wave state measurements.

Contribution

It develops a systematic quantum version of non-linear Radon transforms using star-product quantization, including explicit formulas and the concept of 'thick' quantum tomography.

Findings

Explicit reconstruction formulas for non-linear quantum tomograms.

Introduction of 'thick' quantum tomography with smooth window functions.

Analysis of symmetry groups in generalized tomographic maps.

Abstract

Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear 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 for quantum state measurements of photons and matter waves are discussed.