  We propose a method to infer the single-particle entropy of bosonic atoms in
an optical lattice and to study the local evolution of entropy, spin squeezing,
and entropic inequalities for entanglement detection in such systems. This
method is based on experimentally feasible measurements of
non-nearest-neighbour coherences. We study a specific example of dynamically
controlling atom tunneling between selected sites and show that this could
potentially also improve the metrologically relevant spin squeezing.
