Computing relative entropy in QFT generally requires handling relative modular Hamiltonians, which have nice general properties but are typically very difficult to compute explicitly. Exploiting locality properties of general AQFTs, we describe a scheme that allows to estimate relative modular Hamiltonians from above or below in terms of the modular Hamiltonian of a reference state, which might be better understood. Applying this scheme to coherent states of free scalar fields, the difference between the upper and lower bounds can be made arbitrarily small, recovering an exact result by squeezing. This yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian has not been computed exactly and corroborates the strength of our general estimates. Based on a joint work with Christoph Minz and Ko Sanders. Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006) The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page