In quantum mechanics, subsystems, locality, and entanglement are defined relative to a tensor product structure (TPS) of the Hilbert space, which is extra structure not intrinsic to the Hamiltonian. Cotler, Penington, and Ranard argued that the Hamiltonian's spectrum alone generically determines the local decomposition; Stoica (arXiv:2401.01793, 2024) showed that no spectrum-only method can yield a unique TPS (exponentially many continuous parameters remain free), and that any TPS uniquely determined by the spectrum would be preserved by time evolution, so subsystem entanglement could never change and mutual-information-based emergent distances would be time-degenerate, yielding a frozen 'Carrollian' geometry. The open problem is to characterize, with physical and operational justification, the minimal additional data beyond the spectrum that uniquely singles out a physically meaningful TPS — one supporting entangling dynamics and a non-degenerate emergent spatial geometry — and to establish precise reconstruction conditions in infinite-dimensional Hilbert spaces. Follow-up work has narrowed but not closed this: spectral mereology recovers the number of subsystems but leaves uniqueness open; an operational minimal-scrambling criterion has degenerate cases; a 2025 proof of principle that a Hamiltonian plus a state determine a TPS uses an explicitly non-operational property and is contested by a 2026 no-go comment. The dispute remains unresolved as of 2026.

Background

A quantum theory is usually specified by a Hilbert space H\mathcal H, a Hamiltonian H^\hat H (a Hermitian operator generating time evolution), a state, and a decomposition H=H1⊗H2⊗⋯\mathcal H=\mathcal H_1\otimes\mathcal H_2\otimes\cdots, called the tensor product structure (TPS). The TPS defines subsystems and regions of space: entanglement, subsystem entropies, and local observables are all defined relative to it. The TPS is extra structure — it is not intrinsic to H\mathcal H or to H^\hat H; the only basis-invariant information carried by a Hamiltonian is its spectrum (its eigenvalues with their multiplicities). Zanardi, Lidar, and Lloyd showed that TPSs are observable-induced: a TPS is uniquely fixed by a set of mutually commuting algebras of observables that together generate the full observable algebra.

Hilbert-space fundamentalism (HSF) — the stance underlying the quantum-mereology program of Carroll and Singh — aims to recover all physical structure, including the TPS and three-dimensional space, from minimal quantum ingredients: a Hilbert space, a Hamiltonian, and optionally a state ∣ψ⟩|\psi\rangle. In such programs the factors of the TPS are to be identified with points or discrete building blocks of space, and distances between them are defined by the mutual information (a standard information-theoretic measure of correlation) between the corresponding reduced subsystem states of ∣ψ⟩|\psi\rangle.

The main positive result is due to Cotler, Penington, and Ranard, 'Locality from the Spectrum': if a Hamiltonian is K-local with respect to some TPS — meaning its expansion in single-site operators contains interaction terms involving at most KK subsystems at a time — then, conditional on an unproven existence assumption, almost all such Hamiltonians admit no 'K-dual' (no inequivalent TPS making them K-local), so K-locality selects a unique TPS up to a unitary equivalence acting simultaneously on the pair (H^,TPS)(\hat H,\mathrm{TPS}).

Stoica's three-page comment 'Does the Hamiltonian determine the tensor product structure and the 3d space?' (arXiv:2401.01793) established the obstruction this problem builds on. First, any method that uses only the spectrum must give the same answer in every basis in which H^\hat H is diagonal, hence must be invariant under the full unitary symmetry group of H^\hat H; comparing the dimension of that group with the dimension of the group preserving a TPS shows that the number of continuous parameters left undetermined by the spectrum is exponential in the number of qudits (quantum systems of dd levels each), so no spectrum-only method gives a unique TPS. Second, even if a unique TPS existed, it would have to be preserved by the evolution operators e−iH^te^{-i\hat H t} (which commute with H^\hat H), so a state that starts as a product of subsystem states would remain one: entanglement between subsystems could never change, contrary to observed physics. Third, for emergent-space proposals this means the mutual-information metric would be degenerate along the time direction — the emergent geometry would be 'Carrollian' (a degenerate limit geometry, named after Lewis Carroll, with an absolute time in which nothing moves) — so space would be absolute and everything frozen. Fourth, in infinite-dimensional Hilbert spaces, examples with identical Hamiltonian and unit vector but different TPSs and different numbers of spatial dimensions are known, and even a TPS supplied together with the Hamiltonian and a unit vector is insufficient to describe our world. The paper closes by stating that additional observables, along with their physical meaning, 'are needed to begin with'. The problem posed here is to make that statement precise.

Problem Statement

Characterize the minimal additional data beyond the Hamiltonian's spectrum that uniquely determines a physically meaningful tensor product structure (TPS) of a Hilbert space, together with the operational and physical justification of that data, and the precise reconstruction conditions in infinite-dimensional Hilbert spaces.

Concretely, a constructive resolution must: (1) identify which additional structures, observables, algebras, state-dependent data, or invariants — and with what physical interpretation — must be supplied alongside the spectrum, in a precise sense of 'minimal' (for example, one under which no proper subset of the data singles out a TPS uniquely, and under which the additional data is specifiable without reference to the TPS it is meant to single out, so that data presupposing or encoding that TPS does not qualify), and prove that the spectrum plus this data singles out a TPS uniquely, up to a stated notion of equivalence (for example, local unitaries and permutations of factors, or global-unitary equivalence acting simultaneously on all given structures); (2) justify the choice operationally, so that the selected TPS factors can play the role of subsystems or spatial building blocks: the TPS must be compatible with entangling unitary dynamics (subsystem entanglement entropies must be able to change under evolution by e−iH^te^{-i\hat H t}) and with an emergent spatial geometry whose mutual-information-based distances are capable of being non-degenerate and dynamical rather than frozen into a Carrollian geometry (degenerate along the time direction); and (3) establish, for infinite-dimensional Hilbert spaces, the precise conditions and the necessary supplementary data that permit a faithful reconstruction of subsystem structure and emergent geometry, given the known examples of non-uniqueness in that setting. Alternatively, a rigorous no-go theorem — establishing, under a precise formalization of the physical-relevance conditions in (2), that no additional data of the requested kind can single out a TPS satisfying them, so that observables must be posited as primitive — equally resolves the problem.

The three strands share one objective — a complete, physically justified specification of what beyond the spectrum fixes the subsystem structure — and are jointly linked: the data identified in (1) is exactly what must be operationally justified in (2) and extended in (3).

References

  1. Ovidiu Cristinel Stoica, Does the Hamiltonian determine the tensor product structure and the 3d space?, arXiv:2401.01793 (2024), https://arxiv.org/abs/2401.01793
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