In a solvable, dynamically expanding qubit-tree model with a phase transition between a functioning measurement apparatus and a scrambler (a regime whose dynamics delocalizes information so that no small fraction of it suffices to read that information out), Ferté, Farci and Cao [Phys. Rev. Lett. 136, 090404 (2026)] showed that approximate decoherent histories of a coarse-grained extensive observable emerge in both phases, while objectivity in the sense of quantum Darwinism emerges only in the apparatus phase — a clean separation of two notions of classicality — and that Leggett–Garg inequalities (temporal analogues of Bell inequalities that any macrorealist, non-invasive description must satisfy) are violated at arbitrarily late times by observables close to the model's global Z2\mathbb{Z}_2 symmetry operator, for which the authors offer only a symmetry-based conjecture. They leave open whether this separation persists in conventional lattice many-body models, what dynamical mechanism — plausibly some form of symmetry breaking — stabilizes objectivity under local lattice dynamics, and the precise relation between quantum symmetries and persistent late-time non-classicality. The problem asks to resolve these questions: establish the dynamical conditions under which redundant objective records are stabilized in local many-body dynamics, determine whether the separation between decoherent histories and objectivity is generic there, and settle the symmetry–non-classicality relation.

Background

Two research programs address how classical behavior emerges from quantum mechanics, and they mean different things by "classical". The decoherence program notes that a quantum system interacting with an environment of many degrees of freedom rapidly loses coherence between certain preferred "pointer states" — the states that are least perturbed by the system–environment interaction and that the environment continuously monitors. Its refinement, quantum Darwinism (Ollivier, Poulin and Zurek, 2004), defines classicality as objectivity: information about the system is encoded redundantly, so that many observers measuring disjoint fragments of the environment independently agree on the outcome. Quantitatively, objectivity is diagnosed by a plateau of the quantum mutual information I(S:F)I(S:F) between the system SS and an environment fraction FF as the fraction size varies; a stronger structural condition, the spectrum broadcast structure, requires the joint state to be diagonal in the system's pointer basis with perfectly distinguishable conditional states of different environment fragments; an intermediate strengthening, strong quantum Darwinism, additionally demands that the mutual-information plateau saturate the Holevo bound (the maximum classical information a fragment can carry about SS), so that the conditional fragment states are effectively distinguishable. These formalizations and their relations are reviewed by Korbicz (2021). The decoherent-histories approach instead characterizes classicality intrinsically for an isolated system: one asks whether the unitary evolution supports an ensemble of alternative histories — sequences of outcomes of measurements at successive times — whose quantum amplitudes do not interfere — the interference between any pair of histories, encoded in the off-diagonal elements of the decoherence functional (the matrix of amplitudes connecting different histories), being negligible — so that the dynamics is well approximated by a classical stochastic process over histories. Operationally, approximate decoherent histories are probed by non-disturbance: weakly and coarsely measuring an observable at selected times must not change the outcome statistics that would be obtained at other times, in the way that ideal measurements of a genuinely classical quantity do not disturb it.

A folklore belief held that these two notions of classicality are tied — that decoherent histories require objectivity. This belief was challenged by the Strasberg line of work (2023–2025): large-scale numerics and random-matrix arguments show that slowly varying, coarse-grained extensive observables (sums of local operators) in chaotic, isolated many-body systems admit approximate decoherent histories, even though chaotic dynamics scrambles information — the opposite of redundant accessibility. What was missing was a setting in which objectivity can be switched on and off while decoherent histories are monitored in both regimes. The solvable tree model of Ferté and Cao (2024) provides exactly such a setting. It starts from a qubit apparatus entangled with a measured qubit SS and applies "inflationary" concatenation dynamics: at each time step, every existing apparatus qubit passes through an isometry (a norm-preserving linear map) v(⋅)v(\cdot) producing two qubits, so that after tt steps the apparatus has Nt=2tN_t=2^t qubits while SS is untouched. The tuning angle θ\theta drives a transition at θc=π/4\theta_c=\pi/4 between an apparatus phase that redundantly encodes the state of SS (a quantum-Darwinism plateau: each small fraction of the apparatus suffices to infer SS) and an encoding (scrambler) phase in which information about SS is delocalized into global correlations and inaccessible to any small-fraction measurement.

Ferté, Farci and Cao (2026) then characterized decoherent histories in this model by weakly monitoring the total magnetization Qt=∑jZjQ_t=\sum_j Z_j (sum of Pauli-ZZ operators over apparatus qubits) at fixed relative precision. Their central result is that approximate decoherent histories emerge in both phases once the apparatus is macroscopic (measurements start at time τ≫1\tau\gg 1), with the crucial ingredient being coarse-graining rather than objectivity. The two phases are nevertheless distinguished by the statistics of the histories and their relation to SS: in the encoding phase the history is an ergodic, short-correlated Gaussian process carrying no information about SS; in the apparatus phase the monitored outcome freezes at a run-dependent random value (the histories are non-ergodic), super-observers measuring at different times agree on what the apparatus registered (temporal consensus, in the sense of Touil, Yan and Zurek), and the ensemble of conditional states of SS forms an imperfect pointer-state ensemble that approaches superselection of the pointer basis as θ→0\theta\to 0. Decoherent histories and objectivity are thereby cleanly separated: coarse-grained classicality of histories is generic, while objectivity is the special macroscopic correlation realized in a functioning measurement. The model also retains a non-classical core: the Leggett–Garg inequality — the bound Ct1t2+Ct2t3+Ct3t4−Ct1t4≤2C_{t_1t_2}+C_{t_2t_3}+C_{t_3t_4}-C_{t_1t_4}\le 2 on two-time correlation functions of a dichotomic observable, satisfied by any macrorealist (non-invasive, realist) description — is violated at arbitrarily late times in both phases by global product observables, i.e. tensor powers (aX+bY+cZ)⊗2t(aX+bY+cZ)^{\otimes 2^{t}} of a single-site operator over all apparatus qubits, chosen close to the model's global Z2\mathbb{Z}_2 symmetry operator, and the authors conjecture that the symmetry is responsible without establishing the relation.

What is known about stabilizing objectivity in conventional interacting systems is mostly negative or suggestive. Redundancy established at one time can be destroyed by the environment's own intrinsic dynamics — the "rise and fall of redundancy" of Riedel, Zurek and Zwolak (2012). Two-body interactions between environment constituents drastically limit redundancy (Duruisseau, Touil and Deffner, 2023). Collision-model studies exhibit objectivity only in engineered sequential interactions (Campbell et al., 2019). In the spin-1/2 XX chain (a one-dimensional spin model with nearest-neighbour exchange couplings) prepared in its ground state, the ability to redundantly acquire classical information about the system depends on the quantum phase, and the ferromagnetic (ordered) phase is the only one that completes quantum Darwinism (Giorgi, Galve and Zambrini, 2015) — an existing hint that pointer selection is tied to symmetry breaking. Relatedly, coarse-grained measurement is known to suppress Leggett–Garg violations in generic systems (Kofler and Brukner, 2007), which sharpens the question of why coarse-grained decoherent histories here coexist with persistent violations. Against this background, the source paper conjectures that stabilizing objectivity in conventional lattice dynamics "plausibly requires some form of symmetry breaking" and poses, without resolving: the confrontation of objectivity and decoherent histories in conventional lattice models beyond small sizes; the precise stabilization mechanism and conditions; and the precise relation between quantum symmetries and late-time non-classicality. It also flags the implications of the demonstrated separation for the emergence of classicality in cosmology (reviewed critically by Berjon, Okon and Sudarsky, 2021) as awaiting exploration.

Problem Statement

Establish a dynamical account of the emergence and stability of objectivity in conventional local lattice many-body systems, jointly comprising three linked deliverables that together resolve what the solvable expanding-tree model could only demonstrate in a specially engineered architecture:

(1) Lattice confrontation. Demonstrate — analytically, by controlled large-scale numerics at sizes beyond current small-scale studies, or experimentally in a quantum-simulation platform — a conventional lattice many-body model (local Hamiltonian or local circuit dynamics) that realizes, in different parameter regimes or phases: (a) approximate decoherent histories of a coarse-grained extensive observable (in the operational non-disturbance sense: weak coarse-grained monitoring at selected times does not alter outcome statistics at other times, up to controlled tolerance) together with objectivity in the quantum-Darwinism sense (redundant records about the system accessible to small fractions of the apparatus/environment); and (b) approximate decoherent histories without objectivity. The records counted under (a) must be generated by the lattice's own intrinsic interacting dynamics, not by an independent-site collision/copy coupling to the system or a fine-tuned initial state. Either an explicit model realizing both (a) and (b), or a proof that no conventional local lattice model realizes them jointly, resolves this deliverable — reproducing or refuting, in a realistic local architecture, the separation between the two notions of classicality exhibited by the tree model. The resolution must state whether the separation is generic or confined to fine-tuned models within a specified family of conventional lattice models. The formalization of objectivity used (mutual-information plateau, spectrum broadcast structure, or strong quantum Darwinism) and of approximate decoherent histories must be stated explicitly.

(2) Stabilization mechanism. Identify the dynamical conditions under which local lattice dynamics stabilizes objectivity — in particular settle whether spontaneous symmetry breaking (of the kind that selects an ordered phase and a pointer basis, as suggested by the phase-dependent quantum Darwinism in the XX chain) is necessary, sufficient, or neither; alternatively establish another mechanism with precise hypotheses and scope; or prove that generically no such stabilization is possible. "Stabilize" means that redundant objective records persist at late times under the intrinsic interacting dynamics of the lattice itself, not merely transiently after a fine-tuned or hierarchically arranged initial interaction.

(3) Symmetry–non-classicality relation. Settle the precise relation between residual quantum symmetries and persistent late-time temporal non-classicality: prove or refute that the arbitrarily-late-time Leggett–Garg violations by observables close to a global symmetry operator (such as the tree model's Z2\mathbb{Z}_2 symmetry) are symmetry-protected, i.e. enforced by the unbroken symmetry so that they persist as long as it holds — in the expanding-tree model itself and/or in the lattice setting of (1) — and characterize which symmetries do and do not support such violations alongside coarse-grained decoherent histories.

These deliverables jointly support one overarching objective: a dynamical theory of which many-body systems become classical-with-objective-records (functioning apparatuses) versus merely classical-in-histories (scramblers). A complete resolution requires all three deliverables; an answer addressing only one advances the joint objective but does not resolve the task. The source paper additionally flags implications of the demonstrated separation for the emergence of classicality in cosmology; that direction lies outside the scope of this task.

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