  The performance of beyond mean field methods in solving the quantum many body
problem for fermions is usually characterized by the correlation energy
measured with respect to the underlying mean field value. In this paper we
address the issue of characterizing the amount of correlations associated to
different approximations from a quantum information perspective. With this goal
in mind, we analyze the traditional Hartree-Fock (HF) method with spontaneous
symmetry breaking, the HF with symmetry restoration and the generator
coordinate method (GCM) in a exactly solvable fermion model known as the
three-level Lipkin model. To characterize correlations including entanglement
and beyond we use the quantum discord between different partition orbitals. We
find that for physically motivated partitions, the quantum discord of the exact
ground state is reasonably well reproduced by the different approximations.
However, other partitions create "fake quantum correlations" in order to
capture quantum correlations corresponding to partitions for which the
Hartree-Fock solution fails. Those are removed and redistributed through a
symmetry restoration process.
