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How do symmetries induce natural and useful quantum structures? This questionis investigated in the context of models of three interacting particles inone-dimension. Such models display a wide spectrum of possibilities fordynamical systems, from integrability to hard chaos. This article demonstratesthat the related but distinct notions of integrability, separability, andsolvability identify meaningful collective observables for Hamiltonians withsufficient symmetry. In turn, these observables induce tensor productstructures on the Hilbert space that are especially useful for storing andprocessing quantum information and provide guidance to interpretation andphenomenology of quantum few-body physics.

Symmetry and Natural Quantum Structures for Three-Particles in One-Dimension