  Using the quantum map formalism, we provide a framework to construct fuzzy
and coarse grained quantum states of many-body systems that account for
limitations in the resolution of real measurement devices probing them. The
first set of maps handles particle-indexing errors, while the second deals with
the effects of detectors that can only resolve a fraction of the system
constituents. We apply these maps to a spin-$1/2$ XX-chain obtaining a blurred
picture of the entanglement generation and propagation in the system. By
construction, both maps are simply related via a partial trace, which allow us
to concentrate on the properties of the former. We fully characterize the fuzzy
map, identifying its symmetries and invariants spaces. We show that the volume
of the tomographically accessible states decreases at a double exponential rate
in the number of particles, imposing severe bounds to the ability to read and
use information of a many-body quantum system.
