One-qubit reduced states of a pure many-qubit state: polygon   inequalities
  We show that a necessary and sufficient condition for a set of $n$ one-qubit mixed states to be the reduced states of a pure $n$-qubit state is that their smaller eigenvalues should satisfy polygon inequalities: no one of them can exceed the sum of the others. 
A. Higuchi, A. Sudbery and J. Szulc