  The $N$-representability problem for non-collinear spin-polarized densities
was left open in the pioneering work of von Barth and Hedin setting up the
Kohn-Sham density functional theory for magnetic compounds. In this letter, we
demonstrate that, contrarily to the non-polarized case, the sets of pure and
mixed state $N$-representable densities are different in general. We provide a
simple characterization of the latter by means of easily checkable necessary
and sufficient conditions on the components $\rho^{\alpha \beta} (\br)$ of the
spin-polarized density.
