  Using local density correlation functions for a one-dimensional spin system,
we introduce a correlation function difference (CFD) which compares
correlations on a given site between a full system of size $L$ and its
restriction to $\ell<L$ sites. We show that CFD provides useful information on
transfer of information in quantum many-body systems by considering the
examples of ergodic, Anderson, and many-body localized regimes in disordered
XXZ spin chain. In the ergodic phase, we find that the propagation of CFD is
asymptotically faster than the spin transport but slower than the ballistic
propagation implied by the Lieb-Robinson bound. In contrast, in the localized
cases, we unravel an exponentially slow relaxation of CFD. Connections between
CFD and other observables detecting non-local correlations in the system are
discussed.
