  In a non-equilibrium many-body system, the quantum information dynamics
between non-complementary regions is a crucial feature to understand the local
relaxation towards statistical ensembles. Unfortunately, its characterization
is a formidable task, as non-complementary parts are generally in a mixed
state. We show that for integrable systems, this quantum information dynamics
can be quantitatively understood within the quasiparticle picture for the
entanglement spreading. Precisely, we provide an exact prediction for the time
evolution of the logarithmic negativity after a quench. In the space-time
scaling limit of long times and large subsystems, the negativity becomes
proportional to the R\'enyi mutual information with R\'enyi index $\alpha=1/2$.
We provide robust numerical evidence for the validity of our results for
free-fermion and free-boson models, but our framework applies to any
interacting integrable system.
