Carroll and Singh (Phys. Rev. A 103, 022213 (2021)) proposed to derive, rather than assume, the division of a finite-dimensional Hilbert space into a 'system' and its 'environment': factorizations are ranked by a Schwinger entropy — the maximum of the initial growth rates of the system's entanglement with the environment and of the spreading of its candidate pointer states — and quasi-classical subsystems are those living in factorizations that minimize it. The construction was demonstrated only for a fixed bipartite split, in the interaction-dominated quantum measurement limit, by an impractical random search over factorizations. Later work has produced variational searches for pointer-admitting factorizations (Adil et al., 2026), spectral methods for recursively partitioning systems into weakly interacting sectors (Loizeau and Sels, 2025), an alternative operational objective based on information scrambling (Zanardi et al., 2024), and uniqueness results for preferred tensor product structures that remain disputed (Soulas et al. 2025; Stoica's comment 2026) — but no efficient algorithm for the stated objective, no treatment of the self-Hamiltonian-dominated quantum decoherence limit, and no analysis of the stability of quasi-classical factorizations under perturbation of the factorization. The problem of making the derivation of quasi-classical subsystems from Hamiltonian-level data efficient, general, and robust remains open.

Background

In textbook quantum mechanics the division of a Hilbert space H\mathcal{H} into subsystems — a tensor product structure (TPS), H≅HA⊗HB\mathcal{H}\cong\mathcal{H}_A\otimes\mathcal{H}_B — is fixed before the dynamics is written down. Decoherence theory explains why subsystems described this way can behave classically: environmental monitoring dynamically selects pointer states, states of the system that are robust against entanglement growth with the environment and evolve predictably, a mechanism called einselection (Zurek, Rev. Mod. Phys. 75, 715 (2003)). The program, however, presupposes the split; Zurek himself flagged the identification of 'the systems' from, say, the overall Hamiltonian as a foundational gap, and Zanardi, Lidar, and Lloyd (Phys. Rev. Lett. 92, 060402 (2004)) showed that tensor product structures are induced by the choice of observable algebra rather than being canonical.

Carroll and Singh (Phys. Rev. A 103, 022213 (2021)) inverted the usual logic. Starting from a finite-dimensional Hilbert space equipped with nothing but a Hamiltonian H^\hat H, they asked which of the vastly many factorizations of H\mathcal{H} make the system quasi-classical. Their criteria were: (i) robustness — pointer states should not become rapidly entangled with the environment, quantified by the linear entropy Slin=1−Tr(ρA2)S_{\mathrm{lin}}=1-\mathrm{Tr}(\rho_A^2) of the reduced state; and (ii) predictability — pointer observables should remain localized around approximately classical trajectories, quantified by a 'pointer entropy' SpointerS_{\mathrm{pointer}} measuring the spreading of a wavepacket peaked on the candidate pointer observable. Both are evaluated through second time derivatives at t=0t=0 for initially peaked states and combined into the Schwinger entropy, S¨Schwinger=max⁡(S¨lin(0),S¨pointer(0))\ddot S_{\mathrm{Schwinger}}=\max(\ddot S_{\mathrm{lin}}(0),\ddot S_{\mathrm{pointer}}(0)) (their Eq. (30)), so named because its construction in finite dimensions uses Schwinger's unitary operator basis; its minimization defines the quasi-classical (QC) factorization. Within each trial factorization one must first construct the candidate pointer observable (CPO), the best tensor-product observable compatible with the interaction Hamiltonian. Decoherence theory distinguishes two dynamical regimes: the quantum measurement limit (QML), where the system–environment interaction dominates and pointer states are selected by the interaction — the regime the paper treats — and the quantum decoherence limit (QDL), where the self-Hamiltonian dominates and pointer states are energy eigenstates (Paz and Zurek, Phys. Rev. Lett. 82, 5181 (1999)), which the paper did not work out. The proof of principle recovered, by minimizing the Schwinger entropy over randomly perturbed factorization unitaries, the position-coupled factorization of a finite-dimensional two-oscillator model with fixed subsystem dimensions.

The construction is motivated by the question of whether locality, and ultimately spacetime structure, can be reconstructed from Hilbert-space-level data (Cao, Carroll, and Michalakis, Phys. Rev. D 95, 024031 (2017); Carroll and Singh's 'Mad-Dog Everettianism' program), a question on which Cotler, Penington, and Ranard (Commun. Math. Phys. 368, 1267 (2019)) showed that the Hamiltonian spectrum alone almost never determines a local structure, although dual local descriptions, when they exist, are essentially unique. The expected robustness of the selected factorizations is tied to redundancy — multiple subsystems each recording information about the others — the central mechanism of quantum Darwinism (Riedel, Zurek, and Zwolak, New J. Phys. 14, 083010 (2012)). The paper's Discussion section explicitly lists what remains open: practical feasibility of the search and refinement of the Schwinger-entropy objective; extension to further factorization into multiple subsystems with varying sub-factor dimensions; explicit treatment of the QDL; and whether quasi-classicality survives infinitesimal perturbations of the factorization or is finely tuned. Emergent spacetime and the reconstruction of locality appear in the paper as motivation for the program rather than as one of these open questions.

Problem Statement

Given a finite-dimensional Hilbert space with a Hamiltonian H^\hat H (optionally also a quantum state), and no pre-supposed subsystem structure, complete the quantum-mereology construction of Carroll and Singh by resolving the questions their proof-of-principle leaves open. The single overarching objective is to make the derivation of quasi-classical subsystems from Hamiltonian-level data efficient, general, and robust; the following three facets jointly serve that objective and must be covered by any claimed full resolution. Each facet bundles several distinct deliverables, and a full resolution must address every deliverable named within each of the three facets; addressing a proper subset — of the three facets, or of the deliverables within a single facet — resolves those sub-questions alone and does not by itself constitute a full resolution.

  1. (Efficiency) Provide a computationally feasible algorithm — approximate or heuristic if accompanied by an explicit reliability characterization — for locating low-Schwinger-entropy factorizations, including the nested candidate-pointer-observable optimization, beyond random-walk exploration at proof-of-principle scale; establish whatever convergence or complexity guarantees are possible, or prove they are unavailable.

  2. (Generality) Extend the construction beyond one fixed bipartite split: select subsystem dimensions from the data — each sub-factor of dimension at least two, with the sub-factor dimensions multiplying to dim⁡H\dim\mathcal{H}, so that the trivial one-factor and one-dimensional-factor decompositions are excluded — and define how the quasi-classicality criterion composes over multipartite or hierarchical mereologies; and treat the quantum decoherence limit, where the self-Hamiltonian dominates and pointer states are energy eigenstates, adapting the objective wherever the measurement-limit construction (candidate pointer observable from the interaction Hamiltonian) loses its footing.

  3. (Robustness) Determine whether the quasi-classicality of selected factorizations is stable under infinitesimal perturbations of the factorization unitary or finely tuned: quantify the degradation, characterize the neighborhoods of good factorizations, and settle whether robustness is enhanced by redundant recording across multiple subsystems in the sense of quantum Darwinism.

Answers may refine or replace the Schwinger-entropy objective, provided the replacement is derived from stated physical principles and reduces to the source's criteria on its examples.

In addition, the source motivates the construction by the emergent-spacetime program: demonstrating or refuting that the factorizations selected by the (possibly refined) criterion reproduce familiar locality structure and semiclassical dynamics in models where spatial locality is independently defined would provide valuable physical validation, but the paper poses this as motivation rather than one of its open questions, so it is not a required facet of a full resolution.

References

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