Inferring subsystem structure from accessible dynamics

In finite-dimensional many-body quantum mechanics, entanglement and locality depend on the chosen tensor-product structure; observable algebras provide a more general formulation. The same Hilbert space can admit inequivalent subsystem choices. The inverse question is which choices are identifiable from accessible observables and dynamics, rather than from an assumed preferred factorization.

What is known: The key insight (Zanardi; Zanardi–Lidar–Lloyd) is that a tensor-product structure is not fundamental but is induced by a choice of physically accessible observables — an algebra of operators — and by the dynamics. Entanglement is thus observable-relative: a state can be entangled with respect to one factorization and product with respect to another. The dynamics (locality of the Hamiltonian) singles out preferred factorizations, and there are proposals that the correct subsystem decomposition is the one making the Hamiltonian local ('the Hamiltonian selects the tensor structure', Cotler–Penington–Ranard; Carroll–Singh 'Mad-dog Everettianism'). In algebraic QFT the analogue is the choice of local von Neumann algebras (QF8, QF9).

Once accessible commuting observable algebras are specified, observable-induced subsystems already give a principled construction. A uniquely selected factorization need not exist for arbitrary Hamiltonians, and an observer-independent canonical split is not assumed here.

The remaining inverse problem is to state what partial operational data suffice to recover an approximately local structure, identify ambiguities, and bound sensitivity to experimental or model error. In field and gauge theories, local algebras need not correspond to finite-dimensional tensor factors; QF8/QF9 track that separate setting. Related: QF8, QF9, E15.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.