  Out-of-time ordered (OTO) correlation functions describe scrambling of
information in correlated quantum matter. They are of particular interest in
incoherent quantum systems lacking well defined quasi-particles. Thus far, it
is largely elusive how OTO correlators spread in incoherent systems with
diffusive transport governed by a few globally conserved quantities. Here, we
study the dynamical response of such a system using high-performance
matrix-product-operator techniques. Specifically, we consider the
non-integrable, one-dimensional Bose-Hubbard model in the incoherent
high-temperature regime. Our system exhibits diffusive dynamics in time-ordered
correlators of globally conserved quantities, whereas OTO correlators display a
ballistic, light-cone spreading of quantum information. The slowest process in
the global thermalization of the system is thus diffusive, yet information
spreading is not inhibited by such slow dynamics. We furthermore develop an
experimentally feasible protocol to overcome some challenges faced by existing
proposals and to probe time-ordered and OTO correlation functions. Our study
opens new avenues for both the theoretical and experimental exploration of
thermalization and information scrambling dynamics.
