  We consider scattering of Faddeev-Kulish electrons in QED and study the
entanglement between the hard and soft particles in the final state at the
perturbative level. The soft photon spectrum naturally splits into two parts:
i) soft photons with energies less than a characteristic infrared scale $E_d$
present in the clouds accompanying the asymptotic charged particles, and ii)
sufficiently low energy photons with energies greater than $E_d$, comprising
the soft part of the emitted radiation. We construct the density matrix
associated with tracing over the radiative soft photons and calculate the
entanglement entropy perturbatively. We find that the entanglement entropy is
free of any infrared divergences order by order in perturbation theory. On the
other hand infrared divergences in the perturbative expansion for the
entanglement entropy appear upon tracing over the entire spectrum of soft
photons, including those in the clouds. To leading order the entanglement
entropy is set by the square of the Fock basis amplitude for real single soft
photon emission, which leads to a logarithmic infrared divergence when
integrated over the photon momentum. We argue that the infrared divergences in
the entanglement entropy (per particle flux per unit time) in this latter case
persist to all orders in perturbation theory in the infinite volume limit.
