  Results are presented for the medium-induced, soft coherent radiation
spectrum for all $2\to 2$ partonic channels in QCD, at leading-order in
$\alpha_s$ but beyond leading logarithmic accuracy. The general formula is
valid in the full kinematic range of the underlying process, and reduces to
previous results in special cases. The soft gluon radiation spectrum is
expressed in terms of the color density matrix specific to each channel,
quantifying the entanglement between the color components of the $2 \to 2$
production amplitude. Beyond the leading logarithm, the spectrum depends
explicitly on the off-diagonal elements of this matrix, owing to the soft
gluon's ability to probe the internal color structure of the parton pair.
