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 ∆ of any density matrix ρ. Therefore, a general approach to the
global properties of mixed states depends on the specific metric defined on ∆. In the present work
we shall employ a simple measure on ∆ 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/T r(ρ2) and the
concomitant maximum eigenvalue λm. 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 λm-domain