We consider scattering processes in perturbative QED and compute the entanglement entropy between the hard and the soft particles in the final state. The leading perturbative entanglement entropy diverges logarithmically with respect to the IR cutoff. The coefficient of the divergence is proportional to the cusp anomalous dimension in QED, irrespective of the precise details of the initial state. For two-electron scattering processes, the computations can be extended to all orders in perturbation theory. The Renyi entropies (per unit time, per particle flux) turn out to be proportional to the total inclusive cross-section in the initial state, and so they are free of any IR divergences. Nonetheless, the entanglement entropy exhibits non-analyticity with respect to the IR cutoff. This logarithmic behavior is induced by the small eigenvalues of the hard density matrix, which scale inversely proportional with the size of the box providing the IR cutoff.