Extremal entanglement and mixedness in continuous variable systems

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 $\mathrm{tr}\phantom{\rule{0.3em}{0ex}}{\ensuremath{\varrho}}^{2}$ for the state $\ensuremath{\varrho}$) 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…