  The entanglement of assistance quantifies the entanglement that can be
generated between two parties, Alice and Bob, given assistance from a third
party, Charlie, when the three share a tripartite state and where the
assistance consists of Charlie initially performing a measurement on his share
and communicating the result to Alice and Bob through a one-way classical
channel. We argue that if this quantity is to be considered an operational
measure of entanglement, then it must be understood to be a tripartite rather
than a bipartite measure. We compare it with a distinct tripartite measure that
quantifies the entanglement that can be generated between Alice and Bob when
they are allowed to make use of a two-way classical channel with Charlie. We
show that the latter quantity, which we call the entanglement of collaboration,
can be greater than the entanglement of assistance. This demonstrates that the
entanglement of assistance (considered as a tripartite measure of
entanglement), and its multipartite generalizations such as the localizable
entanglement, are not entanglement monotones, thereby undermining their
operational significance.
