  Local Convertibility refers to the possibility of transforming a given state
into a target one, just by means of LOCC with respect to a given bipartition of
the system and it is possible if and only if all the Renyi-entropies of the
initial state are smaller than those of the target state. We apply this concept
to adiabatic evolutions and ask whether they can be rendered through LOCC in
the sense above. We argue that a lack of differential local convertibility
(dLC) signals a higher computational power of the system's quantum phase, which
is also usually connected with the existence of long-range entanglement,
topological order, or edge-states. Remarkably, dLC can detect these global
properties already by considering small subsystems. Moreover, we connect dLC to
spontaneous symmetry breaking by arguing that states with finite order
parameters must be the most classical ones and thus be locally convertible.
