  We present a necessary and sufficient condition for three qutrit density
matrices to be the one-particle reduced density matrices of a pure three-qutrit
quantum state. The condition consists of seven classes of inequalities
satisfied by the eigenvalues of these matrices. One of them is a generalization
of a known inequality for the qubit case. Some of these inequalities are proved
algebraically whereas the proof of the others uses the fact that a continuous
function of the state must have a minimum. Construction of states satisfying
these inequalities relies on a representation of the convex set of the allowed
set of eigenvalues in terms of corner points. We also present a result for a
more general quantum system concerning the nature of the boundary surface of
the set of the allowed set of eigenvalues.
