Certifying the promotion of finite-time decoherence to stable classicality: a general criterion for einselected histories in equilibrated and energy-eigenstate quantum systems, and the scope of the Eigenstate Einselection Hypothesis

Albrecht, Baunach and Arrasmith (Phys. Rev. D 106, 123507 (2022)) showed in a finite-dimensional adapted Caldeira–Leggett model that einselection — the environment-driven selection of robust pointer states — operates inside a globally equilibrated state and even in single energy eigenstates, and analyzed it with the consistent-histories decoherence functional. They left open the certification problem: the formalism treats a set of histories as classical when its off-diagonal interference terms are 'sufficiently small', but no general, physically motivated quantitative criterion fixes how small that must be or how the tolerance should depend on the intended use of the classical description (e.g., the accuracy demanded of a clock); and their proposed Eigenstate Einselection Hypothesis (EEH), the einselection analogue of the Eigenstate Thermalization Hypothesis, has been tested only in that toy model. Later work shows substantial partial progress — chaos- and size-dependence of interference suppression, and explicit demonstrations that passing consistency, quantum-Darwinism and Leggett–Garg tests does not prevent later recoherence — but supplies no general criterion and no EEH test beyond the toy model. The problem is open: supply the tolerance measure, its use-dependence, and structural stability conditions ensuring that finite-time suppression of interference is not later undone by recoherence, together with the consequences for EEH beyond the toy model, or prove that no such criterion is possible.

Background

How the classical world emerges from quantum theory is usually framed through decoherence: when a quantum system interacts with an environment, superpositions in the basis monitored by the environment lose mutual phase coherence, and only special robust 'pointer states' survive monitoring. The dynamical selection of these preferred states is called einselection. In a closed system with no external observer, probabilities can instead be assigned to coarse-grained time sequences of events ('histories') using the consistent-histories formalism introduced by Griffiths and developed by Gell-Mann and Hartle. For projectors $P_i$ applied at times $t_0$ and $t_1$, one constructs branch states and the decoherence functional $D_{ij,kl}$, whose diagonal entries are candidate probabilities and whose off-diagonal entries measure interference between alternative histories. A set of histories is consistent (decoheres) when the off-diagonal entries are small enough that the diagonal entries obey the sum rules of classical probability; since Dowker and Kent's critique it has been known that many inequivalent approximately consistent sets exist for the same state (the set-selection problem) and that the formalism itself does not say how small 'small' must be.

The source paper connects this to cosmology. Our observed Universe has a thermodynamic arrow of time rooted in a low-entropy initial state, and equilibrium cosmologies — for example, models treating a positive cosmological constant as defining a de Sitter equilibrium state — face tuning puzzles such as the Boltzmann Brain problem, the worry that in an equilibrium universe observers arising as random small fluctuations would vastly outnumber observers like us that evolved from a low-entropy past. A prior objection to such models was that classicality itself requires an arrow of time: if detailed balance makes entangling and disentangling processes equally likely, einselection might fail. The authors tested this in the adapted Caldeira–Leggett (ACL) model, a finite-dimensional analogue of quantum Brownian motion in which a truncated harmonic-oscillator 'system' (dimension 30) couples linearly to a random-Hermitian-matrix 'environment' (dimension 600), the global equilibrium state $|E\rangle$ being a snapshot taken deep in the equilibrated regime. Using predictability-sieve diagnostics (ranking candidate states by how long conditional predictability survives environmental monitoring) and two-time consistent histories, they found that einselection — coherent states remaining stable far longer than position or energy eigenstates — persists inside the equilibrium state; that detailed balance appears instead as a diversity of mutually incompatible decohering history sets, including sets with double-headed arrows of time whose low-entanglement 'middle' moment can be placed anywhere; and that interference stays suppressed thanks to the same weak coupling that supports classicality, whether or not the global state has an arrow of time.

Two limitations of that paper are load-bearing. First, its consistency check is tolerance-dependent: the authors state explicitly that deciding how large interference effects can be before classicality breaks down requires a measure tied to what the classical description is for (their example: the accuracy required if the oscillator is a clock pendulum), which they place outside their scope. Second, Appendix B proposes the 'Eigenstate Einselection Hypothesis' (EEH) — the analogue, for einselection, of the Eigenstate Thermalization Hypothesis (ETH), by which individual energy eigenstates of generic interacting many-body systems already encode thermal behavior (Deutsch; Srednicki). In the ACL model, single eigenstates of the total Hamiltonian that significantly overlap $|E\rangle$ reproduce the einselection features, while eigenstates with little overlap show anomalous, sometimes overwhelming interference; ETH itself concerns many-body field theories, so EEH's domain of validity beyond the toy model is unknown. The question also engages a live dispute: Boddy, Carroll and Pollack argue that in a truly stationary state 'nothing happens' (no dynamical process of observation occurs), a claim the ACL results and Lloyd's decoherent-histories analysis of the cosmological measure problem dispute.

Problem Statement

The problem is to supply — or to prove impossible — a general, physically motivated certification criterion that promotes the finite-time suppression of interference in an isolated quantum system to a stable classical sector, applicable when the global state is an equilibrated snapshot, a typical equilibrated pure state, or a single energy eigenstate. In the criterion form, the answer must specify, and support with nontrivial demonstrations: (i) a quantitative measure of residual interference among histories — normalized off-diagonal entries of the decoherence functional, or an operationally equivalent object — together with an explicit tolerance rule for when a set of histories may be assigned classical probabilities, where the rule depends explicitly on the intended use of the classical description (so that, for example, an oscillator used as a clock pendulum is certified only to the accuracy that use demands); (ii) structural conditions on the Hamiltonian and its spectrum, the global state, and the coarse-grained projectors under which a certification is stable, meaning that subsequent unitary evolution and benign control operations (interventions of the kind shown by Valladares Duque–Strasberg to recohere the 'classical' dynamics of an otherwise decohering toy model) cannot restore interference among the certified histories to a level at which the criterion itself would withhold certification, or destroy the records they leave; and (iii) the consequences for the Eigenstate Einselection Hypothesis: whether these conditions hold for single energy eigenstates of many-body or field-theoretic systems beyond the finite-dimensional adapted Caldeira–Leggett model, and for which eigenstates of the total Hamiltonian. The objective is discharged by either of the following complete answer forms: a validated criterion (a formal statement meeting (i)–(iii) plus worked demonstrations, at least one beyond the adapted Caldeira–Leggett model); or a no-go result (the class of criteria meeting the minimal adequacy condition that a criterion be computable from the Hamiltonian and its spectrum, the global state, and the coarse-grained projectors, and that it withhold certification for the recohering toy model of Valladares Duque–Strasberg; a proof that no criterion in that class can meet the certification requirements (i)–(ii); an explicit finite-dimensional witness; here (iii) lapses, since a no-go against (i)–(ii) leaves no criterion whose EEH consequences could be assessed). A counterexample class to the Eigenstate Einselection Hypothesis — an explicit larger system in which single energy eigenstates fail to show the einselection features found in the adapted Caldeira–Leggett model — would constitute substantial partial progress on (iii), but does not by itself discharge the objective, since it supplies neither the tolerance measure and use-dependence rule of (i) nor the stability conditions of (ii).

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