  Any set of pure states living in an given Hilbert space possesses a natural
and unique metric --the Haar measure-- on the group $U(N)$ of unitary matrices.
However, there is no specific measure induced on the set of eigenvalues
$\Delta$ of any density matrix $\rho$. Therefore, a general approach to the
global properties of mixed states depends on the specific metric defined on
$\Delta$. In the present work we shall employ a simple measure on $\Delta$ that
has the advantage of possessing a clear geometric visualization whenever
discussing how arbitrary states are distributed according to some measure of
mixedness. The degree of mixture will be that of the participation ratio
$R=1/Tr(\rho^2)$ and the concomitant maximum eigenvalue $\lambda_m$. The cases
studied will be the qubit-qubit system and the qubit-qutrit system, whereas
some discussion will be made on higher-dimensional bipartite cases in both the
$R$-domain and the $\lambda_m$-domain.
