The decoherence program explains emergent classicality through einselection — the selection of pointer states robust to entanglement — but it presupposes a tensor product structure (TPS) splitting the Hilbert space into system and environment. Adil, Rudolph, Arrasmith, Holmes, Albrecht and Sornborger (2024, published in Physical Review D 2026) showed that a single finite-dimensional Hamiltonian, given only by its spectrum, generally admits many tensor product structures whose subsystems support pointer states — coexisting quasiclassical 'realms', some strongly interacting and non-local — and posed the resulting global analog of the preferred-basis problem: no physically justified measure or selection principle on the space of realms exists that would determine which factorization is physically realized, the algorithm-induced measure being an optimization artifact. The problem is to derive such a principle from the Hamiltonian (and, if admitted, the global state) — invariant under global unitaries, local unitaries, and factor permutations, and free of built-in classical prejudices such as locality of pointer states or interaction-minimality — and to determine or weight the realized realm or realms; or to prove that no such principle is definable from that data alone. Later literature (2024–2026) supplies contested proof-of-principle uniqueness results and adjacent quantum-mereology constructions but no accepted resolution; the problem remains open.

Background

A composite quantum system is described by a Hilbert space together with a tensor product structure (TPS): a decomposition H≅Hs⊗He\mathcal{H}\cong\mathcal{H}_s\otimes\mathcal{H}_e into factors, each of which describes a subsystem. Entanglement, locality, and the very distinction between a 'system' and its 'environment' are all defined relative to this factorization. Decoherence is the loss of quantum coherence between superposed states of a subsystem caused by entanglement with other degrees of freedom (see Schlosshauer, 2019, for a review). Environment-induced superselection, or einselection, is the stronger dynamical effect (Zurek, 1981, 2003) by which interaction with the environment selects a family of robust states — pointer states — which remain unentangled (or nearly so) from the environment and evolve predictably. Within a fixed TPS, einselection resolves the preferred-basis problem, i.e. the question of which states of the subsystem behave classically; the predictability sieve (Zurek, 1993) identifies pointer states operationally as those that generate the least entanglement with the environment over time.

The whole construction, however, presupposes the TPS: one must specify how the Hilbert space factors before decoherence can even be discussed. Zurek (1998) flagged precisely this gap, remarking that a compelling explanation of what the 'systems' are — how to define them given the overall Hamiltonian on a Hilbert space — would be most useful to the decoherence program. The top-down study of how a Hilbert space factors into subsystems with quasiclassical dynamics has come to be called quantum mereology. Candidate answers include the autonomy criterion of Carroll and Singh (2021), the operational principle of minimal information scrambling of Zanardi, Dallas, Andreadakis and Lloyd (2024), and the mathematical uniqueness result of Cotler, Penington and Ranard (2019), who proved that requiring the Hamiltonian to be KK-local (i.e. to involve interactions among at most KK factors at a time) generically determines a TPS uniquely up to natural equivalences. Stoica (2022, 2024) proved complementary impossibility results: from the Hamiltonian alone, and in a strict sense even together with the state, a unique TPS cannot be selected, because any construction still leaves an abundance of distinct structures related by symmetries.

Adil, Rudolph, Arrasmith, Holmes, Albrecht and Sornborger (2024; published in Physical Review D, 2026) made the question concrete. Given only a Hamiltonian — fundamentally, a spectrum, since any physical description of the operator is relative to a chosen fiducial factorization — they asked which factorizations admit quasiclassical subsystems in the sense that certain 'pointer' states stay robust to entanglement with their complement. They proved that every Hamiltonian admits at least one such factorization, namely the one in which the global energy eigenvectors are separable (a 'block-diagonal' destination Hamiltonian), and they implemented a variational algorithm that discovers many further factorizations for central-spin-type, tensor-product ('quantum measurement limit'), and fully random Hamiltonians. The algorithm's outputs organize into coexisting quasiclassical 'realms' (a realm is the set of quasiclassical worlds associated with one TPS), several of which are strongly interacting and non-local in the fiducial description. This directly undermines the two most popular heuristics for a preferred factorization — that it should support localized states (Piazza, 2010) or minimize the interaction Hamiltonian (Tegmark, 2015; Zanardi, Dallas, Andreadakis and Lloyd, 2024) — since pointer states can exist precisely in strongly interacting, non-local factorizations. The algorithm's cost function does induce a measure over realms, but the authors argue it is an artifact of the optimization procedure with no evident physical significance. The stakes are sharpest in relative-state (many-worlds) readings of quantum mechanics: if several inequivalent TPSs all support einselection, the branches of the universal wavefunction are not uniquely defined even at the level of the subsystem split — a global analog of the preferred-basis problem, with a kinship to the coexisting coarse-grained 'narratives' of the consistent-histories formalism (a formulation of quantum dynamics in terms of decohering sets of coarse-grained histories). The unresolved target is the preferred-realm problem stated below.

Problem Statement

Given a closed, finite-dimensional quantum system described only by a global Hamiltonian HH — that is, by its spectrum, with no pre-specified tensor product structure, basis, subsystem split, or notion of locality — among the generally many tensor product structures (TPSs) of the Hilbert space that admit quasiclassical subsystems (pointer states robust to entanglement with their complement; the 'realms' of Adil et al.), determine which realm or realms are physically realized. The fundamental datum is HH (equivalently, its spectrum); if a resolution admits the global state ∣ψ⟩|\psi\rangle as additional fundamental data, it must state that choice explicitly, and the same choice of data set then applies to both branches below.

Specifically, either:

(1) Derive a physically justified selection principle or measure on the space of such tensor product structures, playing, at the level of the factorization of Hilbert space, the role that the predictability sieve plays for pointer bases within a fixed factorization and that the Born rule plays for branches of the universal wavefunction. The principle must (i) be definable from the fundamental data — the Hamiltonian, and the global state ∣ψ⟩|\psi\rangle if the state is admitted as data — rather than from classical prejudices such as spatial locality of the pointer states, minimality of the interaction Hamiltonian, or a pre-chosen fiducial factorization; (ii) be invariant under the natural symmetries, namely conjugation of HH by a global unitary and the local unitaries and factor permutations that relate equivalent TPSs — if ∣ψ⟩|\psi\rangle is admitted as data, the global-unitary symmetry acts diagonally on the pair, U:(H,∣ψ⟩)↦(UHU†,U∣ψ⟩)U:(H,|\psi\rangle)\mapsto(UHU^\dagger,U|\psi\rangle); and (iii) yield a nontrivial selection or weighting of realms, compatible with the coexistence structure established by Adil et al.; or

(2) Prove a no-go result: no principle satisfying (i)–(iii) can be defined from the specified data alone.

A complete answer to (1) should also state its consequences for which realm or realms describe the quasiclassical world experienced by observers, and why the selection is not an artifact of the formalism in the way the algorithm-induced cost measure is. (The source paper additionally leaves open, as separable ancillary directions not required for this objective: the dynamical-versus-optimization-artifact status of its 'extended coherence time' solutions, convergence and completeness guarantees for its variational search, extensions to larger Hilbert-space dimensions and multipartite factorizations with variable factor dimensions, and the design of experimental probes for non-local realms.)

References

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