  As a universal theory of physics, quantum mechanics must assign states to
every level of description of a system -- from a full microscopic description,
all the way up to an effective macroscopic characterization -- and also to
describe the interconnections among them. Assuming that we only have a
coarse-grained access to a physical system, here we show how to assign to it a
microscopic description that abides by all macroscopic constraints. In order to
do that, we employ general coarse-graining maps, allowing our approach to be
used even when the split between system and environment is not obvious. As a
by-product, we show how effective nonlinear dynamics can emerge from the linear
quantum evolution, and we readily apply it to a state discrimination task.
