Unnormalized quasi-distributions and tomograms of quantum states

TL;DR

This paper investigates unnormalized quantum tomograms and quasi-distribution functions, establishing conditions for density matrix reconstruction and analyzing specific quantum states that violate normalization, thus impacting quantum state characterization.

Contribution

It introduces conditions for reconstructing density matrices from unnormalized tomograms and quasi-distributions, with detailed examples demonstrating normalization violations.

Findings

Certain quantum states have unnormalized tomograms and quasi-distributions.

Reconstruction of the density matrix is not always possible with these functions.

Explicit examples show violations of normalization conditions.

Abstract

Tomograms and quasi-distribution functions like Wigner, Glauber - Sudarshan $P$- and Husimi $Q$- functions that violate the standard normalization condition are considered. Conditions under which a reconstruction of the density matrix using these tomograms and quasi-distribution functions is possible are obtained. Three different examples of states like the de Broglie plane wave, the Moschinsky shutter problem and the stationary state of the charged particle in the uniform and constant electric field are studied. Their tomograms and quasi-distribution functions expressed in terms of the Dirac delta function, the Airy function and the Fresnel integrals are shown to violate the standard normalization condition and thus the density matrix of the state can not always be reconstructed.