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 tunnelling between selected sites and show that this could potentially also improve the metrologically relevant spin squeezing.