  The incoherent dynamical properties of open quantum systems are generically
attributed to an ongoing correlation between the system and its environment.
Here, we propose a novel way to assess the nature of these system-environment
correlations by examining the system dynamics alone. Our approach is based on
the possibility or impossibility to simulate open-system dynamics with
Hamiltonian ensembles. As we show, such (im)possibility to simulate is closely
linked to the system-environment correlations. We thus define the
nonclassicality of open-system dynamics in terms of the nonexistence of a
Hamiltonian-ensemble simulation. This classifies any nonunital open-system
dynamics as nonclassical. We give examples for open-system dynamics that are
unital and classical, as well as unital and nonclassical.
