  We study the entanglement between soft and hard particles produced in generic
scattering processes in QED. The reduced density matrix for the hard particles,
obtained via tracing over the entire spectrum of soft photons, is shown to have
a large eigenvalue, which governs the behavior of the Renyi entropies and of
the non-analytic part of the entanglement entropy at low orders in perturbation
theory. The leading perturbative entanglement entropy is logarithmically IR
divergent. The coefficient of the IR divergence exhibits certain universality
properties, irrespectively of the dressing of the asymptotic charged particles
and the detailed properties of the initial state. In a certain kinematical
limit, the coefficient is proportional to the cusp anomalous dimension in QED.
For Fock basis computations associated with two-electron scattering, we derive
an exact expression for the large eigenvalue of the density matrix in terms of
hard scattering amplitudes, which is valid at any finite order in perturbation
theory. As a result, the IR logarithmic divergences appearing in the
expressions for the Renyi and entanglement entropies persist at any finite
order of the perturbative expansion. To all orders, however, the IR logarithmic
divergences exponentiate, rendering the large eigenvalue of the density matrix
IR finite. The all-orders Renyi entropies (per unit time, per particle flux),
which are shown to be proportional to the total inclusive cross-section in the
initial state, are also free of IR divergences. The entanglement entropy, on
the other hand, retains non-analytic, logarithmic behavior with respect to the
size of the box (which provides the IR cutoff) even to all orders in
perturbation theory.
