  Starting from a coarse-grained map of a quantum many-body system, we
construct the inverse map that assigns a microscopic state to a coarse-grained
state based on the maximum entropy principle. Assuming unitary evolution in the
microscopic system, we examine the resulting dynamics in the coarse-grained
system using the assignment map. We investigate both a two-qubit system, with
swap and controlled-not gates, and $n$-qubit systems, configured either in an
Ising spin chain or with all-to-all interactions. We demonstrate that these
dynamics exhibit atypical quantum behavior, such as non-linearity and
non-Markovianity. Furthermore, we find that these dynamics depend on the
initial coarse-grained state and establish conditions for general microscopic
dynamics under which linearity is preserved. As the effective dynamics induced
by our coarse-grained description of many-body quantum systems diverge from
conventional quantum behavior, we anticipate that this approach could aid in
describing the quantum-to-classical transition and provide deeper insights into
the effects of coarse-graining on quantum systems.
