  New measures of multipartite entanglement are constructed based on two
definitions of multipartite information and different methods of optimizing
over extensions of the states. One is a generalization of the squashed
entanglement where one takes the mutual information of parties conditioned on
the state's extension and takes the infimum over such extensions. Additivity of
the multipartite squashed entanglement is proved for both versions of the
multipartite information which turn out to be related. The second one is based
on taking classical extensions. This scheme is generalized, which enables to
construct measures of entanglement based on the {\it mixed convex roof} of a
quantity, which in contrast to the standard convex roof method involves
optimization over all decompositions of a density matrix rather than just the
decompositions into pure states. As one of the possible applications of these
results we prove that any multipartite monotone is an upper bound on the amount
of multipartite distillable key. The findings are finally related to analogous
results in classical key agreement.
