  The study of the entanglement properties of systems of N fermions has
attracted considerable interest during the last few years. Various separability
criteria for pure states of N identical fermions have been recently discussed
but, excepting the case of two-fermions systems, these criteria are difficult
to implement and of limited value from the practical point of view. Here we
advance simple necessary and sufficient separability criteria for pure states
of N identical fermions. We found that to be identified as separable a state
has to comply with one single identity involving either the purity or the von
Neumann entropy of the single-particle reduced density matrix. These criteria,
based on the verification of only one identity, are drastically simpler than
the criteria discussed in the recent literature. We also derive two
inequalities verified respectively by the purity and the entropy of the single
particle, reduced density matrix, that lead to natural entanglement measures
for N-fermion pure states. Our present considerations are related to some
classical results from the Hartree-Fock theory, which are here discussed from a
different point of view in order to clarify some important points concerning
the separability of fermionic pure states.
