From free-evolution to tomographic representation

TL;DR

This paper develops a method to derive the quantum tomogram of one- and two-particle systems using the free evolution propagator, enabling analysis of quantum states and entanglement through a probability representation.

Contribution

It introduces a general approach to obtain time-dependent quantum tomograms for various systems, including multi-particle states and non-orthogonal configurations, linking to quantum information measures.

Findings

Derived explicit tomograms for Gaussian wave packets and diffraction phenomena.

Extended the formalism to two-particle systems with non-orthogonal states.

Analyzed entanglement properties using linear entropy in the tomographic framework.

Abstract

We use the free evolution propagator to determine the quantum probability representation (i.e., the general expression of the tomogram) of any one-dimensional system described by a density state. The evolution operator for the considered quantum system is additionally used to establish the corresponding time dependent tomogram. Applications are given for a Gaussian wave packet, the quantum shutter related with the phenomenon of diffraction in time, the double quantum shutter, and a finite potential. A generalisation to describe $N$ particle systems is also presented and, in particular, we find the tomogram associated to the 2 particle case occupying in general non-orthogonal states. In the latter case, for a bipartite quantum system, the entanglement properties are established by considering quantum information concepts such as the linear entropy.