  We discuss the entropic criterion for separability of compound quantum
systems for general non-additive entropic forms based on arbitrary concave
functions $f$. For any separable state, the generalized entropy of the whole
system is shown to be not smaller than that of the subsystems, for any choice
of $f$, providing thus a necessary criterion for separability. Nevertheless,
the criterion is not sufficient and examples of entangled states with the same
property are provided. This entails, in particular, that the conjecture about
the positivity of the conditional Tsallis entropy for all $q$, a more stringent
requirement than the positivity of the conditional von Neumann entropy, is
actually a necessary but not sufficient condition for separability in general.
The direct relation between the entropic criterion and the largest eigenvalues
of the full and reduced density operators of the system is also discussed.
