  In the context of quantifying entanglement we study those functions of a
multipartite state which do not increase under the set of local
transformations. A mathematical characterization of these monotone magnitudes
is presented. They are then related to optimal strategies of conversion of
shared states. More detailed results are presented for pure states of bipartite
systems. It is show that more than one measure are required simultaneously in
order to quantify completely the non-local resources contained in a bipartite
pure state, while examining how this fact does not hold in the so-called
asymptotic limit. Finally, monotonicity under local transformations is proposed
as the only natural requirement for measures of entanglement.
