  We have explained and comprehensively illustrated in Part I that the
generalized Pauli constraints suggest a natural extension of the concept of
active spaces. In the present Part II, we provide rigorous derivations of the
theorems involved therein. This will offer in particular deeper insights into
the underlying mathematical structure and will explain why the saturation of
generalized Pauli constraints implies a specific simplified structure of the
corresponding many-fermion quantum state. Moreover, we extend the results of
Part I to non-fermionic multipartite quantum systems, revealing that extremal
single-body information has always strong implications for the multipartite
quantum state. In that sense, our work also confirms that pinned quantum
systems define new physical entities and the presence of pinnings reflect the
existence of (possibly hidden) ground state symmetries.
