Quantum Mereology : Factorizing Hilbert Space into Sub-Systems

How we talk about quantum systems depends crucially on how Hilbert space is factorized, or equivalently on a set of preferred observables. We tackle the question, given a finite-dimensional Hilbert space and a Hamiltonian, without any additional structure, how does one decompose the Hilbert space into a tensor factorization of sub-systems with quasi-classical behavior? A quasi-classical decomposition has features such as low entropy states resilient to entanglement production, existence of preferred pointer observables robust under evolution, and preserving predictive power while decohering sub-systems relatively quickly. We connect these features with properties of the Hamiltonian, in particular locality, and show that arbitrary factorizations will not exhibit quasi-classicality. We make contact with conjugate operators and point out conditions under which they correspond to classical conjugate variables, characterized by classical dynamics. An algorithm which minimizes an entropy-based quantity sifting through factorizations of Hilbert space to select the quasi-classical one is outlined. We remark on the application of this formalism to the emergence of spacetime from quantum dynamics.