  We explore deformations of finite chains of independent fermions which give
rise to large correlations between their extremes. After a detailed study of
the Su-Schrieffer-Heeger (SSH) model, the trade-off curve between end-to-end
correlations and the energy gap of the chains is obtained using
machine-learning techniques, paying special attention to the scaling behavior
with the chain length. We find that edge-dimerized chains, where the second and
penultimate hoppings are reinforced, are very often close to the optimal
configurations. Our results allow us to conjecture that, given a fixed gap, the
maximal attainable correlation falls exponentially with the system size. Study
of the entanglement entropy and contour of the optimal configurations suggest
that the bulk entanglement pattern is minimally modified from the clean case.
