Variation of entanglement entropy in scattering process

In a scattering process, the final state is determined by an initial state and an S-matrix. We focus on two-particle scattering processes and consider the entanglement between these particles. For two types initial states, i.e., an unentangled state and an entangled one, we calculate perturbatively the change of entanglement entropy from the initial state to the final one. Then we show a few examples in a field theory and in quantum mechanics.