  One way to look for complex behaviours in many-body quantum systems is to let
the number $N$ of degrees of freedom become large and focus upon collective
observables. Mean-field quantities scaling as $1/N$ tend to commute, whence
complexity at the quantum level can only be inherited from complexity at the
classical level. Instead, fluctuations of microscopic observables scale as
$1/\sqrt{N}$ and exhibit collective Bosonic features, typical of a mesoscopic
regime half-way between the quantum one at the microscopic level and the
classical one at the level of macroscopic averages. Here, we consider the
mesoscopic behaviour emerging from an infinite quantum spin chain undergoing a
microscopic dissipative, irreversible dynamics and from global states without
long-range correlations and invariant under lattice translations and dynamics.
We show that, from the fluctuations of one site spin observables whose linear
span is mapped into itself by the dynamics, there emerge bosonic operators
obeying a mesoscopic dissipative dynamics mapping Gaussian states into Gaussian
states. Instead of just depleting quantum correlations because of decoherence
effects, these maps can generate entanglement at the collective, mesoscopic
level, a phenomenon with no classical analogue that embodies a peculiar complex
behaviour at the interface between micro and macro regimes.
