  Open quantum systems are inherently coupled to their environments, which in
turn also obey quantum dynamical rules. By restricting to dissipative dynamics,
here we propose a measure that quantifies how far the environment action on a
system departs from the influence of classical noise fluctuations. It relies on
the lack of commutativity between the initial reservoir state and the
system-environment total Hamiltonian. Independently of the nature of the
dissipative system evolution, Markovian or non-Markovian, the measure can be
written in terms of the dual propagator that defines the evolution of system
operators. The physical meaning and properties of the proposed definition are
discussed in detail and also characterized through different paradigmatic
dissipative Markovian and non-Markovian open quantum dynamics.
