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arXiv:2605.06848

Quantum Darwinism and the quality of Petz recovery

Juha Torvinen, Esko Keski-Vakkuri, Nicola Pranzini

quant-phSubmitted May 7, 2026• Announced May 11, 2026• 0 citations• Revised Jul 3, 2026 (v2)
v1 · May 7, 2026v2 · Jul 3, 2026
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Abstract

According to Quantum Darwinism, system-environment interactions both einselect particular system properties and encode them redundantly in many independent subsets of the environment, called fragments. This redundancy implies that an observer can recover the einselected information by accessing just one such fragment. However, the protocol by which such reconstruction should occur is often left unspecified. Considering a system $Γ$ interacting with a multipartite environment $Ξ$, we investigate whether, and under what conditions, the einselected state of $Γ$ can be recovered from environmental fragments using the Petz recovery map. We show that the fidelity between the system's initial state and the state reconstructed via Petz recovery develops a plateau as a function of the fragment size. Our results are supported by both analytical arguments and numerical simulations of large but tractable models.

Comments
14 pages, 5 figures

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According to Quantum Darwinism, system-environment interactions both einselect particular system properties and encode them redundantly in many independent subsets of the environment, called fragments. This redundancy implies that an observer can recover the einselected information by accessing just one such fragment. However, the protocol by which such reconstruction should occur is often left unspecified. Considering a system ΓΓΓ interacting with a multipartite environment ΞΞΞ, we investigate whether, and under what conditions, the einselected state of ΓΓΓ can be recovered from environmental fragments using the Petz recovery map. We show that the fidelity between the system's initial state and the state reconstructed via Petz recovery develops a plateau as a function of the fragment size. Our results are supported by both analytical arguments and numerical simulations of large but tractable models.
The aim in Chapter 7 is to take into account the role of the means of information transmission on the nature of the states that can be perceived. Our point of departure is the recognition that the information we obtain is acquired by observers who monitor fragments of the same environment that decohered the system, einselecting preferred pointer states in the process. Moreover, we only intercept a fraction of the environment. The only information about the system that can be transmitted by its fraction must have been reproduced in many copies in that environment. This process of amplification limits what can be found out to the states einselected by decoherence. Quantum Darwinism provides a simple and natural explanation of this restriction, and, hence, of the objective existence—the essence of classicality—for the einselected states. This chapter introduces and develops information-theoretic tools and concepts (including, e.g., redundancy) that allow one to explore and characterize correlations and information flows between systems, environments, and observers, and illustrates them on an exactly solvable yet non-trivial model.
A quantum system interacting with a multipartite environment can induce redundant encoding of the information of a system into the environment, which is the essence of quantum Darwinism. At the same time, the environment may scramble the initially localized information about the system. Based on a collision model, we mainly investigate the relationship between information scrambling in an environment and the emergence of quantum Darwinism. Our results show that when the mutual information between the system and environmental fragment is a linear increasing function of the fragment size, the tripartite mutual information (TMI) is zero, which can be proved generally beyond the collision model; when the system exhibits Darwinistic behavior, the TMI is positive (i.e., scrambling does not occur); when we see the behavior of an "encoding" environment, the TMI is negative (i.e., scrambling occurs). Additionally, we give a physical explanation for the above results by considering two simple but illustrative examples. Moreover, depending on the nature of system and environment interactions, it is also shown that the single qubit and two-qubit systems behave differently for the emergence of quantum Darwinism, and hence the scrambling, while their relationship is consistent with the above conclusion.
Quantum Darwinism explains classical objectivity through redundant environmental records of a system's pointer states. We show that these records can be established through an intermediate measurement apparatus without direct system--environment interactions. Using exact analytical solutions of quantum circuit models, we compare system monitoring with apparatus monitoring, where the environment interacts only with the apparatus. During encoding, system--apparatus correlations decrease as system--environment correlations grow. Without noise, both models yield identical accessible pointer information and redundancy, while the discord accounts for their different mutual information. The reliability of these records depends on where noise acts. Apparatus bit flips introduce errors shared by successive records, limiting the information recoverable even from the complete environment. In contrast, larger fragments can mitigate independent noise on environment qubits after encoding. Dephasing in the pointer basis leaves the record information unchanged in the model considered. These results extend the description of quantum Darwinism to indirect measurements and clarify the conditions for reliable environmental records.
Abstract Classical objectivity arises when many independent observers can retrieve the same information about a quantum system without disturbing it. Quantum Darwinism (QD) explains this by treating the environment as a communication channel that redundantly broadcasts information about pointer observables. We develop an operational framework for quantifying this redundancy through the lens of functional information. By introducing a one-sided adequacy criterion based on the Holevo bound, we measure the \emph{abundance} of environment fragments that individually contain nearly all classically accessible pointer information. Using a heterogeneous pure–dephasing model, we sample fragments to extract onset statistics and apply isotonic regression to obtain stable estimates of redundancy and its functional-information counterpart FIQD=log⁡2RδFI_{QD}=\log_2 R_\deltaFIQD​=log2​Rδ​. The results reveal a generic onset–plateau structure: redundancy grows rapidly at early times, then saturates at a capacity-limited ceiling, universally across adequacy tolerances. We further connect FIQDFI_{QD}FIQD​ to thermodynamic bounds, showing that each additional bit of redundancy implies exponentially increasing minimal heat costs for record stabilization. This approach provides a conservative, model-agnostic measure of emergent classicality, suitable for both theoretical analysis and experimental probing.
Quantum error correction (QEC) and Quantum Darwinism describe opposing consequences of system-environment interactions: QEC seeks to preserve logical quantum information, whereas Quantum Darwinism explains the emergence of objective classical information through its proliferation into the environment. Despite their common physical origin, no direct quantitative connection between these paradigms has previously been established. We introduce an exactly solvable block-environment model based on the logical GHZ block of the Shor [[9,1,3]] quantum error-correcting code, collectively coupled to N environment qubits. The logical fidelity, Holevo information, and Darwinistic redundancy are obtained systematically for arbitrary environment size and imperfect recovery efficiency. Eliminating the common decoherence parameter yields an exact tradeoff relating Darwinistic redundancy directly to the post-recovery logical fidelity, demonstrating that the emergence of redundant classical records occurs at the expense of logical quantum information. We further prove a model-independent no-go theorem showing that the logical fidelity exceeds a critical threshold precludes the emergence of Darwinistic redundancy, irrespective of the microscopic Hamiltonian or environment structure. The solvable model saturates this general bound, establishing the first quantitative information-theoretic connection between logical quantum information protection and the emergence of redundant classical records.
"How much information about a system S can one extract from a fragment F of the environment E that decohered it?" is the central question of Quantum Darwinism. To date, most answers relied on the quantum mutual information of SF, or on the Holevo bound on the channel capacity of F to communicate the classical information encoded in S. These are reasonable upper bounds on what is really needed but much harder to calculate-the accessible information in the fragment F about S. We consider a model based on imperfect c-not gates where all the above can be computed, and discuss its implications for the emergence of objective classical reality. We find that all relevant quantities, such as the quantum mutual information as well as various bounds on the accessible information exhibit similar behavior. In the regime relevant for the emergence of objective classical reality this includes scaling independent of the quality of the imperfect c-not gates or the size of E, and even nearly independent of the initial state of S.