  Decoherence and einselection have been effective in explaining several
features of an emergent classical world from an underlying quantum theory.
However, the theory assumes a particular factorization of the global Hilbert
space into constituent system and environment subsystems, as well as specially
constructed Hamiltonians. In this work, we take a systematic approach to
discover, given a fixed Hamiltonian, (potentially) several factorizations (or
tensor product structures) of a global Hilbert space that admit a
quasi-classical description of subsystems in the sense that certain states (the
"pointer states") are robust to entanglement. We show that every Hamiltonian
admits a pointer basis in the factorization where the energy eigenvectors are
separable. Furthermore, we implement an algorithm that allows us to discover a
multitude of factorizations that admit pointer states and use it to explore
these quasi-classical "realms" for both random and structured Hamiltonians. We
also derive several analytical forms that the Hamiltonian may take in such
factorizations, each with its unique set of features. Our approach has several
implications: it enables us to derive the division into quasi-classical
subsystems, demonstrates that decohering subsystems do not necessarily align
with our classical notion of locality, and challenges ideas expressed by some
authors that the propensity of a system to exhibit classical dynamics relies on
minimizing the interaction between subsystems. From a quantum foundations
perspective, these results lead to interesting ramifications for relative-state
interpretations. From a quantum engineering perspective, these results may be
useful in characterizing decoherence free subspaces and other passive error
avoidance protocols.
