  We investigate the relationship between mixedness and entanglement for
Gaussian states of continuous variable systems. We introduce generalized
entropies based on Schatten $p$-norms to quantify the mixedness of a state, and
derive their explicit expressions in terms of symplectic spectra. We compare
the hierarchies of mixedness provided by such measures with the one provided by
the purity (defined as ${\rm tr} \varrho^2$ for the state $\varrho$) for
generic $n$-mode states. We then review the analysis proving the existence of
both maximally and minimally entangled states at given global and marginal
purities, with the entanglement quantified by the logarithmic negativity. Based
on these results, we extend such an analysis to generalized entropies,
introducing and fully characterizing maximally and minimally entangled states
for given global and local generalized entropies. We compare the different
roles played by the purity and by the generalized $p$-entropies in quantifying
the entanglement and the mixedness of continuous variable systems. We introduce
the concept of average logarithmic negativity, showing that it allows a
reliable quantitative estimate of continuous variable entanglement by direct
measurements of global and marginal generalized $p$-entropies.
