  Differently from correlation of classical systems, entanglement of quantum
systems cannot be distributed at will - if one system A is maximally entangled
with another system B, it cannot be entangled at all to a third system C. This
concept, known as the monogamy of entanglement, manifests when the entanglement
of A with a pair BC, can be divided as contributions of entanglement between A
and B and A and C, plus a term \tau_{ABC} involving genuine tripartite
entanglement and so expected to be always positive. A very important measure in
Quantum Information Theory, the Entanglement of Formation (EOF), fails to
satisfy this last requirement. Here we present the reasons for that and show a
set of conditions that an arbitrary pure tripartite state must satisfy for EOF
to become a monogamous measure, ie, for \tau_{ABC} \ge 0. The relation derived
is connected to the discrepancy between quantum and classical correlations,
being \tau_{ABC} negative whenever the quantum correlation prevails over the
classical one. This result is employed to elucidate features of the
distribution of entanglement during a dynamical evolution. It also helps to
relate all monogamous instances of EOF to the Squashed Entanglement, an always
monogamous entanglement measure.
