  We introduce geometric measures of entanglement for indistinguishable
particles, which apply to mixed states, multipartite systems, and arbitrary
dimensions. They are based on generalized (i.e., not necessarily finite) norms
on the set of quantum states and lead to the first necessary and sufficient
computational separability criterion in this general setting. The coherent
approach developed in the paper allows us to compare, in particular,
entanglement for fermionic and distinguishable particles: The entanglement
measure for fermionic particles coincides with the corresponding entanglement
measure for distinguishable particles up to a factor of $k!$ where $k$ is the
number of particles involved. By this result the amount of entanglement
emerging from fermi statistics alone is clearly separated from the overall
amount of entanglement. Finally, our techniques are applied to entanglement
related to Schmidt and Slater numbers.
