Scope and robustness of strong quantum Darwinism: departure regimes from mutual-information quantum Darwinism, objectivity with correlated subenvironments, and scalable diagnostics

Strong Quantum Darwinism (SQD), introduced by Le and Olaya-Castro, requires that the classically accessible information about a system saturate both the quantum mutual information and the system entropy for environmental fragments; combined with strong independence of the subenvironments it was proven equivalent to spectrum broadcast structure, and alone it is necessary and sufficient for state objectivity. The equivalence was established in idealized settings: effectively quantum-nondemolition dynamics, mutually independent environments, and diagnostics that presume full state tomography. This task establishes the boundary of validity of the SQD characterization of objectivity outside that idealized setting, in three linked directions: (i) a quantitative dynamical characterization — proof, refutation, or delimitation — of when SQD departs from traditional quantum Darwinism (the source conjecture: non-negligible self-Hamiltonian, coupling not commuting with it, strong system-environment correlations); (ii) a quantitative theory of how correlated subenvironments can be while objectivity survives, including its operational consequences for observers with finer measurements; (iii) certified, scalable evaluation of the SQD diagnostics (Holevo quantity with measurement optimization, accessible information, discord, the objectivity measure $M^{\mathrm{SQD}}$, distance to the spectrum-broadcast set), with error bounds connecting approximate conditions to approximate structure.

Background

How the objective, classical description of a macroscopic system emerges from the quantum behavior of its microscopic constituents is a foundational problem of open quantum systems. The 'pointer basis' of a system coupled to an environment consists of the states most robust against decoherence (the decay of quantum superpositions into classical mixtures through entanglement with environmental degrees of freedom); other states are destroyed and their information dispersed into the environment. Quantum Darwinism (QD), introduced by Zurek and collaborators (Ollivier, Poulin and Zurek, Phys. Rev. Lett. 93, 220401 (2004)), explains objectivity as redundancy: many disjoint fragments of the environment each carry (nearly) complete information about the pointer state of the system. Its standard operational diagnostic is the quantum mutual information $I(S:\mathcal{F})$ — the von Neumann entropy of the system minus its conditional entropy on a fragment — which is observed to plateau at the system entropy $H(S)$ even for fragments comprising a small fraction of the environment.

The structural counterpart is the spectrum broadcast structure (SBS) of Horodecki, Korbicz and Horodecki (Phys. Rev. A 91, 032122 (2015)): a system-fragments state $\rho_{S\mathcal{F}}=\sum_i p_i |i\rangle\langle i|_S\otimes\rho_i^1\otimes\cdots\otimes\rho_i^F$ whose conditional fragment states are perfectly distinguishable, $\rho_i^k\rho_j^k=0$ for $i\neq j$, so that any observer measuring any fragment reads out the value $i$ without disturbing the state — objectivity in the sense of Bohr non-disturbance (the joint state is unchanged by the measurement, its result aside).

The two frameworks can conflict. In Pleasance and Garraway's structured-environment model (Phys. Rev. A 96, 062105 (2017)) and in the side-by-side comparison of Le and Olaya-Castro (Phys. Rev. A 98, 032103 (2018)), the mutual-information plateau of QD can appear even when the state is not objective. Le and Olaya-Castro resolved this in Phys. Rev. Lett. 122, 010403 (2019) by defining strong quantum Darwinism (SQD): there must exist fragments for which the quantum mutual information, the accessible information $I_{acc}(S:\mathcal{F})$ (the maximum classical mutual information between the outcomes of a measurement on the system and a measurement on the fragment, maximized over the choice of both measurements), and the Holevo quantity $\chi(S^{\Pi}:\mathcal{F})$ (the largest classical information carried by the ensemble of post-measurement fragment states induced by a generalized quantum measurement — a POVM, or positive-operator-valued measure — $\Pi$ on the system) all coincide with $H(S)$. The gap $I-\chi$ is a one-sided quantum discord $\mathcal{D}$ — the share of the correlations that is intrinsically quantum and inaccessible to any measurement on the system; SQD demands it vanish while classical information saturates. The Letter proved that SQD plus strong independence — no correlations between subenvironments conditioned on the system — is equivalent to SBS, and that SQD alone is necessary and sufficient for system-state objectivity. It also introduced a degree-of-objectivity measure $M^{SQD}=(H(S)-\chi+\mathcal{D})/(2H(S))$ and related it to a computable quantity $\eta$ that upper-bounds the trace distance (a standard measure of the distinguishability of quantum states) from the state to the nearest SBS state, a bound due to Mironowicz, Korbicz and Horodecki (Phys. Rev. Lett. 118, 150501 (2017)).

Two post-publication developments sharpen the framework. First, Feller, Roussel, Frérot and Degiovanni (Phys. Rev. Lett. 126, 188901 (2021)) showed by an explicit counterexample on three qutrits (three-level quantum systems) that the Letter's pairwise formulation of strong independence, $I(E_j:E_k|S)=0$ (conditional mutual information between subenvironment pairs), is necessary but not sufficient — genuinely multipartite classical correlations can survive it — and gave the corrected condition: the multipartite mutual information of the conditional fragment states, $I(E_1,\ldots,E_F|S)=0$. In their Reply (Phys. Rev. Lett. 126, 188902 (2021)) Le and Olaya-Castro conceded the mathematical expression was necessary-but-not-sufficient while maintaining that the qualitative definition was used throughout. Korbicz's review (Quantum 5, 571 (2021)) adopts the corrected formulation and proves a generalized SBS theorem: objectivity alone fixes the broadcast state structure, strong independence being an additional, physically motivated idealization. Second, the Letter's own examples (its Supplemental Material, Appendix B) show objectivity can survive without strong independence: subenvironments may be correlated, even entangled, provided each fragment still carries the full classical information — though an observer with a finer POVM can then extract extra information about other subenvironments.

The Letter's equivalence was established within idealizations: models with (effectively) quantum-nondemolition dynamics — a negligible self-Hamiltonian $H_S$ (the Hamiltonian of the system in isolation), or a coupling $H_I$ that commutes with it — initially uncorrelated environments, and diagnostics evaluated by full state tomography (complete reconstruction of the quantum state from many measurement settings). The Letter itself points to the further questions that define the present problem — in its Discussion, its Supplemental Material, and its body text respectively: when does SQD deviate from usual QD; how can the role of strong independence be quantified in realistic, correlated many-body environments; and how can the entropic diagnostics be evaluated in practice, given that they require optimization over system measurements and distances to the SBS set that would otherwise demand optimization over all bases.

Problem Statement

Establish the boundary of validity of the strong-quantum-Darwinism characterization of objectivity outside the idealized setting of quantum-nondemolition dynamics, mutually independent subenvironments, and full state tomography. Three linked questions define concrete completion targets under this single objective.

(i) Departure regimes. Establish, quantify, or refute the conjecture that strong quantum Darwinism deviates from traditional mutual-information quantum Darwinism when the system self-Hamiltonian $H_S$ is non-negligible, the coupling Hamiltonian does not commute with it ($[H_S,H_I]\neq 0$), and strong system-environment correlations are present (the motivating examples also displayed non-Markovian dynamics, i.e. open-system evolution with memory, in which information can flow back from the environment to the system). Concretely: prove sufficient and/or necessary dynamical or structural conditions under which the discord-based gap $I(S:\mathcal{F})-\chi(S^{\Pi}:\mathcal{F})$ persists (equivalently, $M^{SQD}$ stays bounded away from zero) for fragments where the mutual-information plateau holds, connect these conditions to classes of system-environment models, and quantify the size and time dependence of the gap as a function of the dynamical parameters.

(ii) Correlated-environment tolerance. Characterize systematically how correlated subenvironments can be while system objectivity survives: quantitative conditions or bounds interpolating between the exact multipartite strong-independence limit $I(E_1,\ldots,E_F|S)=0$ and the known examples of objective states with correlated, even entangled, subenvironments, including the operational consequences for observers with finer measurement capabilities — when can a finer POVM on one subenvironment extract information about the others, and what does that imply for the objectivity of the reconstructed classical picture.

(iii) Tractable diagnostics. Develop efficient, scalable methods — certified bounds, approximations, algorithms, or experimentally implementable witnesses — to evaluate the SQD diagnostics: the Holevo quantity $\chi(S^{\Pi}:\mathcal{F})$ with its optimization over system measurements, the accessible information $I_{acc}(S:\mathcal{F})$, the discord $\mathcal{D}$, the degree-of-objectivity measure $M^{SQD}$, and the computable bound $\eta$ on the trace distance to the set of spectrum broadcast states, for experimentally relevant systems; together with error bounds connecting approximate satisfaction of the SQD conditions to approximate spectrum broadcast structure (an 'approach theorem' for strong quantum Darwinism).

A complete answer delineates the common boundary — which dynamical regimes and which environment-correlation structures preserve the operational-structural equivalence at the SQD level, and how that boundary can be verified at scale without full tomography. The three questions are linked by this shared objective and by the SQD framework itself; partial answers (a counterexample delimiting the conjecture of (i), a correlation-tolerance bound for (ii), or a certified scalable diagnostic for (iii)) are meaningful components of the same program rather than separate problems.

References

1. Thao P. Le and Alexandra Olaya-Castro, 'Strong Quantum Darwinism and Strong Independence are Equivalent to Spectrum Broadcast Structure', Phys. Rev. Lett. 122, 010403 (2019), DOI 10.1103/PhysRevLett.122.010403, arXiv:1803.08936, https://doi.org/10.1103/PhysRevLett.122.010403
2. Thao P. Le and Alexandra Olaya-Castro, 'Objectivity (or lack thereof): Comparison between predictions of quantum Darwinism and spectrum broadcast structure', Phys. Rev. A 98, 032103 (2018), DOI 10.1103/PhysRevA.98.032103, https://doi.org/10.1103/PhysRevA.98.032103
3. Graeme Pleasance and Barry M. Garraway, 'Application of quantum Darwinism to a structured environment', Phys. Rev. A 96, 062105 (2017), DOI 10.1103/PhysRevA.96.062105, https://doi.org/10.1103/PhysRevA.96.062105
4. R. Horodecki, J. K. Korbicz, and P. Horodecki, 'Quantum origins of objectivity', Phys. Rev. A 91, 032122 (2015), DOI 10.1103/PhysRevA.91.032122, https://doi.org/10.1103/PhysRevA.91.032122
5. Harold Ollivier, David Poulin, and Wojciech H. Zurek, 'Objective Properties from Subjective Quantum States: Environment as a Witness', Phys. Rev. Lett. 93, 220401 (2004), DOI 10.1103/PhysRevLett.93.220401, https://doi.org/10.1103/PhysRevLett.93.220401
6. P. Mironowicz, J. K. Korbicz, and P. Horodecki, 'Monitoring of the Process of System Information Broadcasting in Time', Phys. Rev. Lett. 118, 150501 (2017), DOI 10.1103/PhysRevLett.118.150501, arXiv:1607.02478, https://doi.org/10.1103/PhysRevLett.118.150501
7. Alexandre Feller, Benjamin Roussel, Irénée Frérot, and Pascal Degiovanni, 'Comment on "Strong Quantum Darwinism and Strong Independence Are Equivalent to Spectrum Broadcast Structure"', Phys. Rev. Lett. 126, 188901 (2021), DOI 10.1103/PhysRevLett.126.188901, arXiv:2101.09186, https://doi.org/10.1103/PhysRevLett.126.188901
8. Thao P. Le and Alexandra Olaya-Castro, 'Reply to Comment on "Strong Quantum Darwinism and Strong Independence are Equivalent to Spectrum Broadcast Structure"', Phys. Rev. Lett. 126, 188902 (2021), DOI 10.1103/PhysRevLett.126.188902, arXiv:2101.10756, https://doi.org/10.1103/PhysRevLett.126.188902
9. J. K. Korbicz, 'Roads to objectivity: Quantum Darwinism, Spectrum Broadcast Structures, and Strong quantum Darwinism — a review', Quantum 5, 571 (2021), DOI 10.22331/q-2021-11-08-571, arXiv:2007.04276, https://doi.org/10.22331/q-2021-11-08-571
10. Thao P. Le and Alexandra Olaya-Castro, 'Witnessing non-objectivity in the framework of strong quantum Darwinism', Quantum Sci. Technol. 5, 045012 (2020), DOI 10.1088/2058-9565/abac4e, arXiv:1908.08818, https://doi.org/10.1088/2058-9565/abac4e
11. Diana A. Chisholm, Guillermo García-Pérez, Matteo A. C. Rossi, Sabrina Maniscalco, and G. Massimo Palma, 'Witnessing objectivity on a quantum computer', Quantum Sci. Technol. 7, 015022 (2022), DOI 10.1088/2058-9565/ac40f3, arXiv:2110.06243, https://doi.org/10.1088/2058-9565/ac40f3
12. Davide Girolami, Akram Touil, Bin Yan, Sebastian Deffner, and Wojciech H. Zurek, 'Redundantly Amplified Information Suppresses Quantum Correlations in Many-Body Systems', Phys. Rev. Lett. 129, 010401 (2022), DOI 10.1103/PhysRevLett.129.010401, arXiv:2202.09328, https://doi.org/10.1103/PhysRevLett.129.010401
13. Diana A. Chisholm, Luca Innocenti, and G. Massimo Palma, 'The meaning of redundancy and consensus in quantum objectivity', Quantum 7, 1074 (2023), DOI 10.22331/q-2023-08-03-1074, arXiv:2211.09150, https://doi.org/10.22331/q-2023-08-03-1074
14. Alexandre Feller, Benjamin Roussel, Adrien Pontlevy, and Pascal Degiovanni, 'Testing quantum Darwinism dependence on observers' resources', arXiv:2306.14745 (2023), https://arxiv.org/abs/2306.14745
15. Fernando Galve, Roberta Zambrini, and Sabrina Maniscalco, 'Non-Markovianity hinders Quantum Darwinism', Sci. Rep. 6, 19607 (2016), DOI 10.1038/srep19607, https://doi.org/10.1038/srep19607
16. Nadia Milazzo, Salvatore Lorenzo, Mauro Paternostro, and G. Massimo Palma, 'Role of information backflow in the emergence of quantum Darwinism', Phys. Rev. A 100, 012101 (2019), DOI 10.1103/PhysRevA.100.012101, arXiv:1901.05826, https://doi.org/10.1103/PhysRevA.100.012101
17. Eoghan Ryan, Eoin Carolan, Steve Campbell, and Mauro Paternostro, 'Commutativity and the emergence of classical objectivity', J. Phys. Commun. 6, 095005 (2022), DOI 10.1088/2399-6528/ac8f19, arXiv:2206.08249, https://doi.org/10.1088/2399-6528/ac8f19
18. Nina Megier, Andrea Smirne, Steve Campbell, and Bassano Vacchini, 'Correlations, information backflow, and objectivity in a class of pure dephasing models', Entropy 24, 304 (2022), DOI 10.3390/e24020304, arXiv:2201.10573, https://doi.org/10.3390/e24020304
19. Xiao-Kan Guo and Zhiqiang Huang, 'On the relation between quantum Darwinism and approximate quantum Markovianity', Phys. Lett. A 491, 129204 (2023), DOI 10.1016/j.physleta.2023.129204, https://doi.org/10.1016/j.physleta.2023.129204
20. Emery Doucet and Sebastian Deffner, 'Classifying Two-Body Hamiltonians for Quantum Darwinism', Phys. Rev. X 14, 041064 (2024), DOI 10.1103/PhysRevX.14.041064, arXiv:2405.00805, https://doi.org/10.1103/PhysRevX.14.041064
21. Emery Doucet and Sebastian Deffner, 'Compatibility of Quantum Measurements and the Emergence of Classical Objectivity', arXiv:2411.10862 (2024), https://arxiv.org/abs/2411.10862
22. Diana A. Chisholm, G. Massimo Palma, and Luca Innocenti, 'On the emergence of quantum Darwinism and pointer states for non-commuting evolutions', arXiv:2510.06867 (2025), https://arxiv.org/abs/2510.06867
23. Irene Valladares Duque and Philipp Strasberg, 'Toy model challenging prevailing definitions of classicality', EPL 152, 28001 (2025), DOI 10.1209/0295-5075/ae1639, arXiv:2412.08444, https://doi.org/10.1209/0295-5075/ae1639
24. Anthony Kiely, Diana A. Chisholm, Akram Touil, Sebastian Deffner, Gabriel Landi, and Steve Campbell, 'Metrological approach to the emergence of classical objectivity', Phys. Rev. A 113, 022403 (2026), DOI 10.1103/hn78-7xx3, arXiv:2510.12313, https://doi.org/10.1103/hn78-7xx3
25. David Rebollo-Martínez and Manuel Rebollo-Salas, 'Hidden Coherence Bursts and Time-Tagged Holevo Fragment Information Under Finite-Resolution Observation in a Central-Spin Environment', Entropy 28, 842 (2026), DOI 10.3390/e28080842, https://doi.org/10.3390/e28080842
26. Marin Girard, Gong Cheng, and ChunJun Cao, 'Demystifying Objectivity with Operator Algebra Quantum Error Correction', arXiv:2606.06588 (2026), https://arxiv.org/abs/2606.06588