  How do symmetries induce natural and useful quantum structures? This question
is investigated in the context of models of three interacting particles in
one-dimension. Such models display a wide spectrum of possibilities for
dynamical systems, from integrability to hard chaos. This article demonstrates
that the related but distinct notions of integrability, separability, and
solvability identify meaningful collective observables for Hamiltonians with
sufficient symmetry. In turn, these observables induce tensor product
structures on the Hilbert space that are especially useful for storing and
processing quantum information and provide guidance to interpretation and
phenomenology of quantum few-body physics.
