A geometric approach to the distribution of quantum states in bipartite physical systems

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.