  Euclidean volume ratios characterizing the typicality of entangled and
separable states are investigated for two-qubit and qubit-qutrit quantum
states. For this purpose a new numerical approach is developed. It is based on
the Peres-Horodecki criterion, on a characterization of the convex set of
quantum states by inequalities resulting from Newton identities and Descartes'
rule of signs and on combining this characterization with standard and
Multiphase Monte Carlo algorithms. Our approach confirms not only recent
results on two-qubit states but also allows for a numerically reliable
numerical treatment of so far unexplored special classes of two-qubit and
qubit-qutrit states. However, our results also hint at the limits of efficiency
of our numerical Monte Carlo approaches which is already marked by the most
general qubit-qutrit states forming a convex set in a linear manifold of
thirtyfive dimensions.
