  Gaussian bipartite states are basic tools for the realization of quantum
information protocols with continuous variables. Their complete
characterization is obtained by the reconstruction of the corresponding
covariance matrix. Here we describe in details and experimentally demonstrate a
robust and reliable method to fully characterize bipartite optical Gaussian
states by means of a single homodyne detector. We have successfully applied our
method to the bipartite states generated by a sub-threshold type-II optical
parametric oscillator which produces a pair of thermal cross-polarized
entangled CW frequency degenerate beams. The method provide a reliable
reconstruction of the covariance matrix and allows to retrieve all the physical
information about the state under investigation. These includes observable
quantities, as energy and squeezing, as well as non observable ones as purity,
entropy and entanglement. Our procedure also includes advanced tests for
Gaussianity of the state and, overall, represents a powerful tool to study
bipartite Gaussian state from the generation stage to the detection one.
