Robust spectrum broadcast structure: quantitative error bounds, partial decoherence, and experimental verification of quantum objectivity

Horodecki, Korbicz, and Horodecki [Phys. Rev. A 91, 032122 (2015)] proved that, under full decoherence, Bohr non-disturbance, and strong independence of the observed environment fragments, objective existence of a quantum system's state is equivalent to a spectrum broadcast structure (SBS). They explicitly left open whether the entropic quantum-Darwinism plateau $I[\varrho_{S:fE}]=H_S$ implies SBS ('We leave this, in general uneasy, question open for further research'), called strong independence 'in a sense an idealization', proved dynamical SBS formation only for the illuminated-sphere model with pure environments in the thermodynamic and deep-decoherence limits, and suggested laboratory verification only 'in principle' via quantum state tomography. Later work answered the entropic-sufficiency question: strong quantum Darwinism (vanishing one-sided discord plus accessible-information saturation per fragment) together with a corrected strong-independence condition is equivalent to SBS, and for globally pure states the branching structure is the unique zero-discord structure. Dynamical formation has been established in several model classes with mixed environments. What remains open is the robust core of the program: (A) a universal approximate SBS theory — the missing 'universal approach theorem' allowing errors in the strong-quantum-Darwinism conditions and bounding the resulting departure from ideal SBS, beyond the interaction classes covered so far; (B) a rigorous treatment of partial decoherence and higher-dimensional pointer sectors — the regime the source framework excluded by assumption — with an operational notion of partial objectivity; (C) direct experimental certification of approximate SBS, beyond the mutual-information plateau, in a natural decoherence process.

Background

Decoherence theory explains how a quantum system $S$ interacting with an environment $E$ loses quantum coherence in a preferred 'pointer basis' picked out by the interaction, but it does not by itself explain why the resulting classical information is redundant — why many observers can independently and repeatedly read out the same value. Quantum Darwinism (QD) addresses this: for an environment split into fragments $E_1,\dots,E_N$, it studies the quantum mutual information $I[\varrho_{S:fE}]$ between $S$ and fractions $fE$ of the environment, and in particular the 'plateau' regime $I[\varrho_{S:fE}]=H_S$ (for every fraction $f$), where $H_S$ is the entropy of the decohered system state, meaning each sufficiently large fragment alone carries (nearly) full information about $S$ (Ollivier, Poulin, and Zurek, Phys. Rev. Lett. 93, 220401 (2004); Zurek, Nat. Phys. 5, 181 (2009); demonstrated for scattered photons by Riedel and Zurek, Phys. Rev. Lett. 105, 020404 (2010)).

Horodecki, Korbicz, and Horodecki (Phys. Rev. A 91, 032122 (2015); the source of this problem) gave this picture a state-level foundation. They define objectivity operationally: a state of $S$ exists objectively if many observers can independently, and without disturbing (in Bohr's sense of non-disturbance: leaving the joint state and all future measurement statistics unchanged), find out the same state of $S$. Their main theorem states that, assuming full decoherence of $S$ into a pointer basis $\{|i\rangle\}$ and strong independence of the observed fragments (the only correlations between fragments are those induced by the common system value), objectivity is equivalent to the joint state having a spectrum broadcast structure (SBS): $\varrho_{S:fE}=\sum_i p_i\,|i\rangle\langle i|_S\otimes\varrho^1_i\otimes\cdots\otimes\varrho^k_i,\quad \varrho^k_i\,\varrho^k_{i'}=0\ \text{for}\ i\neq i',$ i.e. the environment factors into conditionally independent records whose supports perfectly distinguish the pointer value. SBS states imply the entropic plateau, but the converse was left open: the paper exhibits a two-qubit entangled state (a mixture of two orthogonal entangled pure states) with $I[\varrho_{S:E}]=H_S$ yet entanglement verified by the PPT (positive partial transpose) criterion, and explicitly leaves the general question open. The paper also acknowledged: strong independence is an idealization; the dynamical proof of SBS formation was given only for the illuminated-sphere model (a sphere scattering an environment of photons) with pure environments in thermodynamic and deep-decoherence limits; and laboratory verification 'in principle' via quantum state tomography.

The subsequent characterization theory is as follows. Quantum discord is the part of the mutual information that is not accessible through local measurements (Ollivier–Poulin–Zurek, above). Writing $\chi(E|S)$ for the Holevo quantity (the maximum, over rank-1 measurements on $S$, of the classical information the environment carries about the outcome) and $I_{\mathrm{acc}}(S:E_k)$ for the accessible information (the maximum, over measurements on $S$, of the classical mutual information with fragment $E_k$), strong quantum Darwinism (SQD) is the conjunction of conditions $I(S:E)=\chi(E|S),\qquad I_{\mathrm{acc}}(S:E_k)=H(S)\ \text{for every}\ k,$ while the corrected strong-independence condition, imposed in addition to SQD, is the vanishing conditional multipartite mutual information $I(E_1\cdots E_N|S)=\sum_k H(E_k|S)-H(E_1\cdots E_N|S)=0.$ Le and Olaya-Castro (Phys. Rev. Lett. 122, 010403 (2019)) proved SBS is equivalent to SQD plus strong independence; Feller, Roussel, Frérot, and Degiovanni (Phys. Rev. Lett. 126, 188901 (2021)) showed by counterexample that pairwise conditional independence is necessary but not sufficient and supplied the corrected joint condition above, which was accepted in the Reply. Korbicz's review (Quantum 5, 571 (2021)) further proved a generalized SBS theorem: objectivity is equivalent to the orthogonal-support broadcast form with no assumption of independence between fragments, so strong independence is not needed for objectivity per se. For globally pure states, Touil, Anzà, Deffner, and Crutchfield (Quantum 8, 1494 (2024)) closed the pure-state side: the branching (SBS) structure is the only structure compatible with zero discord, and small nonzero discord with good decoherence forces the state arbitrarily close to branching.

On the dynamical side, SBS formation has been proven in quantum Brownian motion with mixed Gaussian environments (Tuziemski and Korbicz, EPL 112, 40008 (2015); J. Phys. A 49, 445301 (2016)), the non-Markovian spin-boson model, a two-level system coupled to a bath of harmonic oscillators (Lampo et al., Phys. Rev. A 96, 012120 (2017)), for generic von Neumann measurements with a coarse-grained apparatus and random-matrix genericity of objectivization times (Korbicz et al., Phys. Rev. A 96, 032124 (2017)), with finite-time error control via an 'approach theorem' for pure dephasing (Mironowicz, Korbicz, and Horodecki, Phys. Rev. Lett. 118, 150501 (2017)), for von Neumann-type interaction Hamiltonians and their continuous-variable versions via a Gram–Schmidt sufficient condition (Acevedo, Wehr, and Korbicz, J. Math. Phys. 65, 122102 (2024); Acevedo, Korbicz, and Wehr, J. Math. Phys. 67, 012202 (2026)), for two-body Hamiltonians classified by their support of QD (Doucet and Deffner, Phys. Rev. X 14, 041064 (2024)), and with emergence rates quantified by the quantum Fisher information (a sensitivity measure from quantum metrology) for the spin-star model of one central spin coupled to many environment spins (Kiely et al., Phys. Rev. A 113, 022403 (2026)). However, no error-bound theorem is known beyond the restricted interaction classes covered by these results: the review explicitly identifies as missing a 'universal approach theorem' that would allow error in the SQD conditions and bound the resulting departure from ideal SBS, as well as practically checkable forms of those conditions, whose Holevo and accessibility maximizations are hard to evaluate. The framework also still assumes full decoherence and a maximally resolving direct observation (one-dimensional pointer projectors on $S$); partially decohered systems and higher-dimensional pointer sectors (superselection-like situations) lack an operational notion of partial objectivity and corresponding structure theorems.

Experimentally, by 2019 three experiments had tested only the entropic/Holevo plateau: photonic cluster states (Ciampini et al., Phys. Rev. A 98, 020101(R) (2018)), a photonic quantum simulator (Chen et al., Sci. Bull. 64, 580 (2019)), and nitrogen-vacancy (NV) centers in diamond (Unden et al., Phys. Rev. Lett. 123, 140402 (2019)), the last measuring only the Holevo quantity, which is too weak to infer the state structure. No direct tomographic detection of approximate SBS in a natural decoherence setting has been reported; proposed interfaces include NV centers with dynamically polarized nuclear environments (Kwiatkowski, Cywiński, and Korbicz, New J. Phys. 23, 043036 (2021)), witnessing on NISQ (noisy intermediate-scale quantum) devices of a solvable collision model, in which the system interacts sequentially with individual environment units (Chisholm et al., arXiv:2110.06243), and objectivity-loophole Bell tests on public quantum computers (Bednorz et al., Adv. Quantum Technol. 9, e70420 (2026)).

Problem Statement

The open problem recorded here concerns three unresolved directions, which share one overarching objective — turning the qualitative, exact-regime equivalence between objectivity and spectrum broadcast structure into a quantitative, partially-decoherence-compatible, and experimentally testable theory of objective record formation:

(A) Universal approximate SBS theory (the missing 'universal approach theorem'). Korbicz's review (Quantum 5, 571 (2021)) names the surviving gap: what is missing is 'some universal approach theorem, e.g. of the type [Mironowicz et al.], which would allow for some error in (the strong-QD conditions) and bound the resulting departure from the ideal SBS'. Concretely: if the strong-quantum-Darwinism conditions ($I(S:E)=\chi(E|S)$ and $I_{\mathrm{acc}}(S:E_k)=H(S)$ for every fragment $k$), together with the corrected strong-independence condition ($I(E_1\cdots E_N|S)=0$), hold up to deviations, bound the distance of the joint state from a spectrum broadcast structure in a named distinguishability measure such as trace distance, beyond the classes where such control already exists (the existing approach theorem is proven for pure dephasing; other established formation results cover von Neumann-type measurement interactions and two-body classified Hamiltonians). The bound must be explicit and quantitative: it must state the dependence of the distance on the deviations, vanishing as the deviations vanish, rather than a non-constructive existence or continuity statement. The exact equivalence — SBS if and only if SQD plus corrected strong independence — is already established (Theorem 6 of the review) and is the zero-error reference point rather than a result to be reproved. Related aspects named in the literature are scope description rather than separate mandatory requirements: making the conditions practically evaluable despite their Holevo/accessibility maximizations, and controlling finite decoherence (residual coherences), non-orthogonality of the conditional fragment states, and finite fragment number and size.

(B) Partial decoherence and higher-dimensional pointer sectors. The source framework assumes full decoherence into a pointer basis, explicitly excluding systems that decohere 'in higher-dimensional pointer superselection sectors' because it targets 'the full objectivisation of a given quantum degree of freedom, rather than a partial one'. Extending the theory to that excluded regime is an open direction: an operational definition of partial objectivity for systems that decohere only into higher-dimensional pointer sectors (superselection-like sectors) or whose direct observation is not maximally resolving, together with structure theorems characterizing the state structures compatible with partial objectivity, generalizing the known orthogonal-support broadcast theorem beyond one-dimensional pointer resolution on the system and reducing to it in the full-decoherence limit.

(C) Experimental certification. The source suggested only that 'in a laboratory, this can be in principle directly verified via e.g. quantum state tomography', and the review's conclusions note that experimental investigations of SBS in realistic situations are missing. What has been tested so far is the entropic/Holevo plateau, not the state structure. An open direction is the certification of approximate spectrum broadcast structure in a natural (not state-engineered) decoherence process, by routes such as quantum state tomography or certified witnesses, with robustness to detector noise and imperfect coarse graining quantified. Here certification means an explicit, quantified certification scheme — a protocol or certified witness together with its robustness analysis, applicable to a natural decoherence process; measured data are not required.

The three directions are independent: a complete resolution of any one of (A), (B), or (C) counts as solving this task, and no direction is a prerequisite for another. They are related in content — certification of approximate SBS along (C) would draw on the deviation measures sought in (A), and realistic experiments operate in the partially decohered regime addressed by (B). Note on scope relative to the source: the originally open sufficiency of the entropic plateau and the role of strong independence have been answered by later literature (see Background) and are part of the lineage, not of what remains to be solved.

References

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