  In this job, we will present a theory called Quantum Tomography that is the
natural extension of the theory of detection of signals in classical
telecommunications to Quantum Mechanics. This theory mainly consists in the
reconstruction of a quantum state of a system through a probability
distribution measured directly in the laboratory, usually called Tomogram.
  This Thesis contains five chapters. In the first one, we will show the birth
of this theory as an adaptation of homodyne and heterodyne classical detection
to Quantum Mechanics. In the second, we will describe a tomographic description
of Quantum Mechanics on C*-algebras by splitting the theory in two parts: a
Generalized Sampling Theory and a Generalized Positive Transform. In the third,
we will present the first numerical algorithm that solves the Clebsh-Gordan
decomposition problem for any finite-dimensional unitary representation of any
finite or compact Lie group. This algorithm receives the name of SMILY
algorithm. In the fourth and fifth chapters, we will discuss tomography of
systems parametrized with fields. In the fourth, we will reconstruct states of
classical systems and in the fifth, we will do the same with quantum ones, but
here in two different ways: the first one will be done by adapting the
reconstruction formula of chapter four, and the second one by means of the
axiomatic description of Wightamn-Streater of a Quantum Field Theory.
