  We characterize the information dynamics of strongly disordered systems using
a combination of analytics, exact diagonalization, and matrix product operator
simulations. More specifically, we study the spreading of quantum information
in three different scenarios: thermalizing, Anderson localized, and many-body
localized. We qualitatively distinguish these cases by quantifying the amount
of remnant information in a local region. The nature of the dynamics is further
explored by computing the propagation of mutual information with respect to
varying partitions. Finally, we demonstrate that classical simulability, as
captured by the magnitude of MPO truncation errors, exhibits enhanced
fluctuations near the localization transition, suggesting the possibility of
its use as a diagnostic of the critical point.
