The spectrum broadcast structure (SBS) theorem of Horodecki, Korbicz, and Horodecki identifies the objective existence of a quantum system's state — many observers independently and non-disturbingly determining it — with a classical-quantum branching joint state in which every environment fraction carries a perfectly distinguishable record of the pointer value. All dynamical models that produce SBS, both in the source paper and in nearly all later work, use product-form (measurement-type) interactions whose pointer basis is fixed by the interaction Hamiltonian itself; the source explicitly asks whether more general interaction Hamiltonians, without an a priori chosen pointer basis, can dynamically generate a stable pointer basis and asymptotically form an SBS state. Prior work shows substantial but partial progress: two-body finite-dimensional Hamiltonians supporting quantum Darwinism have been classified (they must admit a time-independent pointer observable satisfying commutation relations, with no system-mediated environment mixing); rigorous dynamical SBS-convergence theorems exist for von Neumann-type measurement interactions including continuous spectra; and non-commuting evolutions have been shown to still allow a relaxed notion of SBS pointer states. The general dynamical-selection problem — proving or refuting, for interaction classes genuinely beyond the commuting-pointer (measurement) form, that dynamics from generic initial states selects a unique stable pointer basis and converges to SBS — remains open.

Background

Decoherence theory explains the emergence of classical behavior through the interaction of a quantum system with its environment: certain states are robust under environmental monitoring and become the preferred basis in which the system's reduced state diagonalizes; this selection process is called einselection and the robust states are called pointer states (Zurek, Reviews of Modern Physics 75, 715 (2003)). Quantum Darwinism (Ollivier, Poulin, and Zurek, Physical Review Letters 93, 220401 (2004); Zurek, Nature Physics 5, 181 (2009)) adds an information-theoretic layer: the environment is divided into independent fragments, and a system property is objective when many observers, each measuring a different fragment, can independently and without disturbance infer the same classical information about the system. Operationally this is detected by a plateau of the quantum mutual information I(S:F)I(S:F) (the von Neumann mutual information between the system and an environment fraction FF) at the system's entropy over a broad range of fragment sizes. Because the quantum mutual information also contains quantum discord — a measure of genuinely quantum, non-classical correlations — the plateau alone does not force a classical state structure.

Horodecki, Korbicz, and Horodecki (arXiv:1305.3247 (2013); journal version R. Horodecki, J. K. Korbicz, P. Horodecki, Physical Review A 91, 032122 (2015); companion Letter Physical Review Letters 112, 120402 (2014)) made this operational notion precise. Starting from a definition of objective existence — many observers can independently find out the state of the system without perturbing it — with non-disturbance understood in Bohr's sense (the statistics of any later measurement are unchanged by the observation), and adding strong independence (conditioned on the system's state, the environment fragments carry no residual mutual correlations), they proved that the only compatible joint states have the spectrum broadcast structure (SBS) ϱS:E=∑ipi∣i⟩⟨i∣⊗kϱiEk,ϱiEkϱi′Ek=0 (i≠i′),\varrho_{S:E}=\sum_i p_i |i\rangle\langle i|\otimes_k \varrho_i^{E_k},\quad \varrho_i^{E_k}\varrho_{i'}^{E_k}=0\ (i\neq i'), where ⊗k\otimes_k denotes a tensor product over environment fragments EkE_k, ∣i⟩|i\rangle are pointer states of the system, and the conditional fragment states ϱiEk\varrho_i^{E_k} have orthogonal supports, i.e. each fragment carries a perfectly distinguishable record of the pointer value ii. SBS implies the quantum Darwinism plateau; the converse was later completed by Le and Olaya-Castro (Physical Review Letters 122, 010403 (2019)), who showed that strong quantum Darwinism (saturation of the accessible information from every fragment, plus vanishing conditional multipartite information) is equivalent to SBS.

The theorem is kinematic: it states which state structure objectivity forces, not which dynamics produces it. All dynamical examples analyzed at the time — the illuminated dielectric sphere, spin-spin, spin-boson, and quantum Brownian motion models — use interaction Hamiltonians of the product form Hint=g AS⊗∑kXEkH_{int}=g\,A_S\otimes\sum_k X_{E_k}, or finite sums ∑iAi⊗Bi\sum_i A_i\otimes B_i of von Neumann measurement type, for which the pointer basis is simply the eigenbasis of ASA_S (or the joint eigenbasis of the commuting family {Ai}\{A_i\}): as the authors put it, the pointer observable is arguably put in by hand by the choice of ASA_S. The initial states are likewise always assumed to be product between system and environment and across environment fractions. The paper therefore closes with an explicit question: are there more general interaction Hamiltonians, without a priori chosen pointer basis, which nevertheless lead to the asymptotic formation of a spectrum broadcast state? Are there truly dynamical mechanisms leading to stable pointer bases and objective classical states? This is the modern structural form of the long-standing pointer-basis selection problem, related to Zurek's predictability sieve — the idea that pointer states are those that minimize entropy production under environmental monitoring. It is compounded by a second assumption of the original derivation: everything is proven for initially uncorrelated (product) environments, so the effect of pre-existing macroscopic correlations or entanglement across observed environment fractions on the possibility and form of spectrum broadcasting is unknown.

Problem Statement

Characterize, by rigorous dynamical criteria, under which system-environment interactions the joint evolution from generic initially uncorrelated (product) states dynamically selects a stable pointer basis and converges asymptotically to a spectrum broadcast structure, when the pointer observable is not fixed a priori by the interaction's commutation algebra. For dynamical selection beyond measurement-form interactions, determine for which interaction Hamiltonians outside the product/measurement forms Hint=g AS⊗∑kXEkH_{int}=g\,A_S\otimes\sum_k X_{E_k} and ∑iAi⊗Bi\sum_i A_i\otimes B_i — for example with [HS,Hint]≠0[H_S,H_{int}]\neq 0, with several mutually non-commuting system operators, with three-body or intra-environmental terms, with time dependence beyond the already classified families, or acting on infinite-dimensional (continuous-spectrum) systems — the joint state of the system and the observed environment macrofractions fEfE converges to ϱS:fE≈∑ipi∣i⟩⟨i∣⊗kϱiEk,ϱiEkϱi′Ek=0 (i≠i′),\varrho_{S:fE}\approx\sum_i p_i |i\rangle\langle i|\otimes_k \varrho_i^{E_k},\quad \varrho_i^{E_k}\varrho_{i'}^{E_k}=0\ (i\neq i'), where convergence is quantified, for example, by vanishing trace distance to the nearest SBS state or by vanishing pairwise fidelities of the conditional environment states in every observed fraction, and where the basis {∣i⟩}\{|i\rangle\} is selected by the dynamics itself — stable, unique for generic initially uncorrelated (product) states, and not merely the eigenbasis of an operator whose commutation with HSH_S and the interaction was built into the Hamiltonian. Both constructive answers (exhibit and prove such mechanisms) and classificatory or impossibility answers (prove that no such mechanism exists within a stated class) are admissible. A complete resolution of the problem is one such result — constructive or impossibility — for a single class of interactions that lies beyond the published two-body finite-dimensional classification (or is otherwise not covered by it); results extending the treatment to further interaction classes are equally admissible, and classes left untreated remain open.

The question is which dynamics produce objective classical states, and it is not settled by the existing literature. Initial states of the system and environment, and across environment fractions, are taken to be uncorrelated (product); initial correlations or entanglement — whether between the system and the environment or across observed environment fractions — are outside the scope of this question.

References

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