  In the spirit of the generalized one-particle density matrix for fermions, we
introduce generalized one- and two-particle density matrices to state
representability conditions up to second order for boson systems without
assuming particle number-conservation. Furthermore, we show for both particle
species that, for a semibounded Hamiltonian, the in?mum of the variation of the
energy functional w.r.t quasifree states coincides with the one of a variation
over pure quasifree states. Moreover, it is proven for fermions that only pure
quasifree states have a generalized 1-pdm that is a projection, and a similar
statement for bosons.
