Soulas-Franzmann-Di Biagio (arXiv:2512.07468) proved that, in finite dimension, a Hamiltonian with non-degenerate spectrum together with a state of full support on its eigenbasis uniquely determines a tensor product structure (TPS) up to local unitaries, via a unitary-invariant property fixing the entanglement entropies of the polynomial orbit R(H^)∣ψ⟩R(\hat{H})|\psi\rangle of the state. The open problem is whether the minimal, 'Hilbert-space-fundamentalist' data (H,H^,∣ψ⟩)(\mathcal{H}, \hat{H}, |\psi\rangle) can determine a tensor product structure in a physically satisfactory way - or whether a rigorous obstruction exists. A satisfactory account must not prespecify the number and dimensions of the subsystems, should extend beyond finite dimension, must use an operationally interpretable and checkable determining property rather than a hand-fixed list of entropy values, and must survive the time-compatibility trilemma raised by Stoica's Comment (2026): under Schrödinger evolution the construction apparently cannot be simultaneously invariant under time translations, unique, and physically relevant. Sharpening sub-questions - which collections of target entropies are realizable (an inverse problem adjacent to the quantum marginal problem) and what minimal data suffice - are open in the source.

Background

In standard quantum mechanics a composite system is described by a tensor product H≅H1⊗⋯⊗Hn\mathcal{H} \cong \mathcal{H}_1 \otimes \cdots \otimes \mathcal{H}_n of subsystem Hilbert spaces, but the Hilbert space H\mathcal{H} by itself does not single out any such factorization: on a finite-dimensional Hilbert space there are infinitely many inequivalent tensor product structures (TPSs) - equivalence classes of isomorphisms H→⨂iHi\mathcal{H} \to \bigotimes_i \mathcal{H}_i, modulo local unitaries U1⊗⋯⊗UnU_1 \otimes \cdots \otimes U_n and permutations of the factors - and which of them deserves to be called 'the subsystems' is normally put in by hand. Quantum mereology is the research program, running from Zanardi-Lidar-Lloyd's observation that tensor product structures are observable-induced (relative to the operationally accessible algebras of interactions and measurements) through Carroll-Singh's search for factorizations with quasi-classical dynamics, that asks which decomposition is preferred by intrinsic, dynamical or operational criteria. Its most austere variant is Hilbert-space fundamentalism (HSF): the position that all physical structure - the subsystem decomposition, locality, and derivatively spacetime - should emerge from minimal quantum ingredients, typically a Hilbert space, a Hamiltonian H^\hat{H}, and a state vector ∣ψ⟩|\psi\rangle.

Two apparently conflicting theorems frame whether this is possible. Cotler, Penington and Ranard ('Locality from the spectrum') showed that if a Hamiltonian on an NN-dimensional Hilbert space is KK-local (its expansion in the operator basis induced by a TPS involves products of at most KK nontrivial single-site operators; the number nn of factors and their dimensions did_i with ∏idi=N\prod_i d_i = N are fixed) in some TPS, then for KK sufficiently small almost all such Hamiltonians (with respect to Lebesgue / Gaussian-unitary-ensemble measure on Hermitian matrices) admit no inequivalent tensor product structure in which they are also KK-local (no 'KK-dual'), so the locality-compatible TPS is unique up to equivalence - a difficult proof relying on algebraic geometry and on an incompletely verified existence assumption. Stoica proved no-go results in the opposite direction: any structure constructed from H^\hat{H} and ψ⟩\psi\rangle alone that is 'physically relevant' - standing in an observable, time-dependent relation to the evolving state - fails to be unique, with the solution space growing exponentially in the number of qudits, and even a unique TPS so obtained would not admit changing subsystem entanglement.

Soulas, Franzmann and Di Biagio dissolved the apparent contradiction (the two results use different notions of uniqueness: 'inequivalent' versus strictly 'distinct') and formalized the physically appropriate question. They introduce kinds - sets carrying an action of the unitary group U(H)U(\mathcal{H}), such as the TPSs with fixed (n;d1,…,dn)(n; d_1,\dots,d_n) or the HSF data pairs with fixed spectrum and support - and call a unitary-invariant property PP on a product kind K0×KeK_0 \times K_e determining if it holds on exactly one U(H)U(\mathcal{H})-orbit (relational uniqueness: structures related by a global unitary give the same predictions and are deemed equivalent). Their central reduction (Proposition 3.5 and Corollary 3.7) reduces determining a product kind to selecting a single orbit of the desired structure under the stabilizer of the fixed input structure - the subgroup of U(H)U(\mathcal{H}) that leaves the input structure unchanged. Their Theorem 3.9 then proves, as a proof of principle, that if H^\hat{H} has non-degenerate spectrum and ∣ψ⟩|\psi\rangle has full support on the eigenbasis, the property PLP_L fixing the entanglement entropies (von Neumann entropies S(ρiT)=−tr(ρiln⁡ρi)S(\rho^{\mathcal{T}}_i) = -\mathrm{tr}(\rho_i \ln \rho_i) of the reduced states of each factor, where ρi\rho_i is obtained by tracing out the other factors) of every vector R(H^)∣ψ⟩R(\hat{H})|\psi\rangle in the polynomial orbit of the state - a countable rational-coefficient subfamily already suffices - determines the product kind of HSF data and TPSs of prespecified type (n;d1,…,dn)(n; d_1,\dots,d_n): for fixed (H^,∣ψ⟩)(\hat{H}, |\psi\rangle) the TPS satisfying PLP_L is unique up to products of single-site unitaries.

Three features keep this from being a physically satisfactory account of subsystem emergence, and they delimit the open problem. (i) The number nn of factors and their dimensions (di)(d_i) must be prespecified; only the arrangement within a fixed factorization type is determined. (ii) Everything is finite-dimensional. (iii) The determining property PLP_L has no direct operational meaning: its list of entropy values is essentially fixed by hand and is not practically checkable. Moreover, in a Comment Stoica has argued that the construction cannot be simultaneously invariant (including under time translations), unique, and physically relevant: under Schrödinger evolution the subsystem entropies of ∣ψ(t)⟩|\psi(t)\rangle change in time, so either the selected TPS depends on an absolute instant, or the invariants must themselves change in time, or entanglement cannot change at all - a trilemma he presents as confirming his no-go theorems. He also observes that (H^,∣ψ⟩)(\hat{H}, |\psi\rangle) already determine a unique eigenbasis (eigenphases fixed by the state), so the entropy conditions merely select among finitely many combinatorial options. Determining a TPS from minimal data is also the prerequisite for programs that would derive spatial or causal structure from a preferred factorization in quantum gravity, where, conversely, gauge constraints and gravitational dressing (the construction of local observables relative to dynamical reference data) are known to obstruct exact Hilbert-space factorization.

Problem Statement

Central question: can the minimal 'Hilbert-space-fundamentalist' data - a Hilbert space, a Hamiltonian H^\hat{H}, and a state vector ∣ψ⟩|\psi\rangle - uniquely determine, in a physically meaningful way, the tensor product structure (TPS) of the Hilbert space; or is there a rigorous obstruction? A complete answer takes one of two forms.

(A) A determination theorem for TPSs from (H^,∣ψ⟩)(\hat{H}, |\psi\rangle) that removes the shortcomings of Soulas-Franzmann-Di Biagio's Theorem 3.9: it must not prespecify the number nn of tensor factors or their dimensions (di)(d_i) - subsystem cardinality and sizes must be outputs of the determination, or the impossibility of achieving this must itself be proven (e.g. a rigorous argument that unitary-invariant data cannot distinguish factorization types); it should extend beyond finite-dimensional Hilbert spaces and bounded operators where possible; the determining property PP must be physically interpretable and operationally checkable rather than a hand-fixed list of entropy values; and the result must settle whether the determined TPS can be simultaneously invariant under the symmetries of the data (in particular time translations), unique, and physically relevant in the sense that subsystem entanglement can still evolve in time - the trilemma raised by Stoica's Comment. For (A) to be well-defined, PP counts as physically interpretable and operationally checkable only if it is stated without reference to a prespecified factorization type and is decidable from finitely many quantities computable from the data (H^,∣ψ⟩)(\hat{H}, |\psi\rangle) alone; entropy-valued conditions remain admissible, provided their target values are derived from the data rather than fixed by hand.

(B) A rigorous impossibility theorem showing that no unitary-invariant property of (H^,∣ψ⟩)(\hat{H}, |\psi\rangle) can select a tensor product structure (or any emergent structure) that is simultaneously unique and physically relevant - i.e. a proof that the trilemma is fatal, strengthening and subsuming Stoica's no-go results within the relational-uniqueness framework.

Sharpening sub-questions, left open by the source: which collections LL of target entropies are realizable as the entanglement data of some TPS and state (an inverse problem related to the quantum marginal problem, which asks which collections of reduced states are compatible with a global pure state); and what is the minimal set of vectors R(H^)∣ψ⟩R(\hat{H})|\psi\rangle whose entanglement data suffice to determine a TPS (a countable set is known to suffice; whether finitely many, or a smaller structured family, can is open). A general-framework companion question posed by the same source: is there a general criterion that, given two kinds K0K_0 and KeK_e, decides whether some unitary-invariant property determines their product kind, and can one characterize the kinds for which no complete set of discriminating invariants exists?

These parts jointly support one objective - a settled answer to whether and under what conditions subsystem structure emerges from the Hamiltonian and the state alone. Partial resolutions of individual facets (e.g. an infinite-dimensional extension with prespecified factors, or an operational determining property with factor count still prespecified) constitute progress but do not by themselves complete the objective; completion is a determination theorem meeting the physical-adequacy conditions of (A) (or the impossibility, admitted within (A), of removing the prespecification), or a decisive impossibility result as in (B).

References

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