  In some many-body systems, certain ground state entanglement (Renyi)
entropies increase even as the correlation length decreases. This entanglement
non-monotonicity is a potential indicator of non-classicality. In this work we
demonstrate that such a phenomenon, known as non-local convertibility, is due
to the edge state (de)construction occurring in the system. To this end, we
employ the example of the Ising chain, displaying an order-disorder quantum
phase transitions. Employing both analytical and numerical methods, we compute
entanglement entropies for various system bipartitions (A|B) and consider
ground states with and without Majorana edge states. We find that the thermal
ground states, enjoying the Hamiltonian symmetries, show non-local
convertibility if either A or B are smaller than, or of the order of, the
correlation length. In contrast, the ordered (symmetry breaking) ground state
is always locally convertible. The edge states behavior explains all these
results and could disclose a paradigm to understand local convertibility in
other quantum phases of matter. The connection we establish between
convertibility and non-local, quantum correlations provides a clear criterion
of which features a universal quantum simulator should possess to outperform a
classical machine.
