On tomographic representation on the plane of the space of Schwartz operators and its dual

TL;DR

This paper explores the mathematical structure of optical quantum tomograms, demonstrating they form a Frechet space, and characterizes the dual space as symbols of quantum observables involving position and momentum polynomials.

Contribution

It establishes the topological structure of quantum tomograms and identifies the dual space as a class of symbols for quantum observables, advancing the mathematical framework of quantum tomography.

Findings

Optical quantum tomograms form a Frechet space.

The dual space consists of symbols of quantum observables.

Includes all polynomials of position and momentum operators.

Abstract

It is shown that the set of optical quantum tomograms can be provided with the topology of Frechet space. In such a case the conjugate space will consist of symbols of quantum observables including all polynomials of the position and momentum operators.