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With the birth of quantum information science, many tools have been developedto deal with many-body quantum systems. Although a complete description of suchsystems is desirable, it will not always be possible to achieve this goal, asthe complexity of such description tends to increase with the number ofparticles. It is thus crucial to build effective quantum theories aiming tounderstand how the description in one scale emerges from the description of adeeper scale. This contribution explores different mathematical tools to thestudy of emergent effective dynamics in scenarios where a system is subject toa unitary evolution and the coarse-grained description of it is given by a CPTPmap taking the original system into an \emph{effective} Hilbert space ofsmaller dimension. We see that a well-defined effective dynamics can only bedefined when some sort of matching between the underlying unitary and thecoarse-graining map is satisfied. Our main goal is to use these different toolsto derive necessary and sufficient conditions for this matching in the generalcase.

Investigating Coarse-Grainings and Emergent Quantum Dynamics with Four Mathematical Perspectives