  We present analytical and numerical studies of the behaviour of the
$\alpha$-Renyi entropies in the Toric code in presence of several types of
perturbations aimed at studying the simulability of these perturbations to the
parent Hamiltonian using local operations and classical communications (LOCC) -
a property called local-convertibility. In particular, the derivatives, with
respect to the perturbation parameter, present different signs for different
values of $\alpha$ within the topological phase. From the information-theoretic
point of view, this means that such ground states cannot be continuously
deformed within the topological phase by means of catalyst assisted local
operations and classical communications (LOCC). Such LOCC differential
convertibility is on the other hand always possible in the trivial disordered
phase. The non-LOCC convertibility is remarkable because it can be computed on
a system whose size is independent of correlation length. This method can
therefore constitute an experimentally feasible witness of topological order.
