  Our goal is to clarify the relation between entanglement and correlation
energy in a bipartite system with infinite dimensional Hilbert space. To this
aim we consider the completely solvable Moshinsky's model of two linearly
coupled harmonic oscillators. Also for small values of the couplings the
entanglement of the ground state is nonlinearly related to the correlation
energy, involving logarithmic or algebraic corrections. Then, looking for
witness observables of the entanglement, we show how to give a physical
interpretation of the correlation energy. In particular, we have proven that
there exists a set of separable states, continuously connected with the
Hartree-Fock state, which may have a larger overlap with the exact ground
state, but also a larger energy expectation value. In this sense, the
correlation energy provides an entanglement gap, i.e. an energy scale, under
which measurements performed on the 1-particle harmonic sub-system can
discriminate the ground state from any other separated state of the system.
However, in order to verify the generality of the procedure, we have compared
the energy distribution cumulants for the 1-particle harmonic sub-system of the
Moshinsky's model with the case of a coupling with a damping Ohmic bath at 0
temperature.
