  The study of conditional $q$-entropies in composite quantum systems has
recently been the focus of considerable interest, particularly in connection
with the problem of separability. The $q$-entropies depend on the density
matrix $\rho$ through the quantity $\omega_q = Tr\rho^q$, and admit as a
particular instance the standard von Neumann entropy in the limit case $q\to
1$. A comprehensive numerical survey of the space of pure and mixed states of
bipartite systems is here performed, in order to determine the volumes in state
space occupied by those states exhibiting various special properties related to
the signs of their conditional $q$-entropies and to their connections with
other separability-related features, including the majorization condition.
Different values of the entropic parameter $q$ are considered, as well as
different values of the dimensions $N_1$ and $N_2$ of the Hilbert spaces
associated with the constituting subsystems. Special emphasis is paid to the
analysis of the monotonicity properties, both as a function of $q$ and as a
function of $N_1$ and $N_2$, of the various entropic functionals considered.
