One-Shot Quantum Darwinism: Entropic Redundancy Thresholds and the Emergence of Classicality via Measured Rényi Mutual Information, with Applications to Trapped Ion Quantum Processors
We introduce a one-shot information-theoretic approach to quantum Darwinism, the process by which classical objectivity arises from quantum mechanics through the redundant encoding of information about a quantum system in a set of independent fragments of an environment. The core information-theoretic object is the measured smooth Renyi-2 mutual information I_epsilon,M_2(S : F)_rho, given by the supremum over the set of local product measurements of the measured smooth Renyi-2 divergence D_epsilon,T_2. This object is given a semidefinite programming (SDP) variational representation analogous to the Regula-Tomamichel formulation of the measured smooth Renyi divergence. Using this object, we prove matching achievability and converse results for the one-shot redundancy function R_epsilon,delta, as well as a classicality criterion. We show that for the i.i.d. fragment model, the condition I_epsilon,M_2(S : F1) >= delta * H_epsilon,M_min(S) is both sufficient for (eps, delta)-objectivity and asymptotically necessary up to a log(1/eps) correction term, which vanishes for eps -> 0. We also prove a second-order asymptotic expansion of the redundancy function, a strong converse exponent for the rate of decay of objectivity below the classicality threshold, and a phase diagram of the quantum Darwinism transition. We apply this formalism to 171Yb+ ion quantum processors, obtaining explicit bounds on the redundancy function for the phonon-mediated environment model in terms of the Lamb-Dicke parameter, decoherence rates, and chain length. All of the results are found to be consistent with the orthonormality of the normal mode eigenvectors.
We introduce a one-shot information-theoretic approach to quantum Darwinism, the process by which classical objectivity arises from quantum mechanics through the redundant encoding of information about a quantum system in a set of independent fragments of an environment. The core information-theoretic object is the measured smooth Renyi-2 mutual information I_epsilon,M_2(S : F)_rho, given by the supremum over the set of local product measurements of the measured smooth Renyi-2 divergence D_epsilon,T_2. This object is given a semidefinite programming (SDP) variational representation analogous to the Regula-Tomamichel formulation of the measured smooth Renyi divergence. Using this object, we prove matching achievability and converse results for the one-shot redundancy function R_epsilon,delta, as well as a classicality criterion. We show that for the i.i.d. fragment model, the condition I_epsilon,M_2(S : F1) >= delta * H_epsilon,M_min(S) is both sufficient for (eps, delta)-objectivity and asymptotically necessary up to a log(1/eps) correction term, which vanishes for eps -> 0. We also prove a second-order asymptotic expansion of the redundancy function, a strong converse exponent for the rate of decay of objectivity below the classicality threshold, and a phase diagram of the quantum Darwinism transition. We apply this formalism to 171Yb+ ion quantum processors, obtaining explicit bounds on the redundancy function for the phonon-mediated environment model in terms of the Lamb-Dicke parameter, decoherence rates, and chain length. All of the results are found to be consistent with the orthonormality of the normal mode eigenvectors.
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=log2Rδ. 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 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.
What does a rising redundancy curve actually prove? This deposit accompanies “Raw-Fragment Benchmarking for Redundancy Tests in Quantum Darwinism and Spectrum Broadcast Structure,” a paper built around a practical but often hidden problem in tests of quantum objectivity. Aggregate mutual-information curves can show that information grows with fragment size, but they do not reveal whether independent fragments really carry accessible, mutually consistent records of the same pointer variable. The missing evidence lives one level lower: in the raw, shot-indexed fragment records. The paper turns that gap into a reproducible benchmark. Starting from a fixed synthetic binary pointer-copy model with 100,000 shots and eight heterogeneous fragments, it asks what a raw-fragment evidence package should expose before redundancy, consensus, or Spectrum Broadcast Structure-style objectivity is inferred. The analysis enumerates all fragment subsets, separates mean-over-subsets and all-subsets threshold crossings, measures fragment disagreement, reproduces analytic binary-channel threshold maps, applies coupling-breaking negative controls, and stress-tests the result against hidden common-mode correlated errors. The resulting picture is sharper than an aggregate curve. In the baseline model, the 0.9 information-fraction threshold is crossed at subset size m=3 under the mean criterion, but only at m=4 under the stricter all-subsets criterion. Negative controls remain below threshold, while a hidden common-mode flip exposes the main fragility axis: many agreeing fragments can still fail as evidence for independent records if they share the same unobserved error source. The attached PDF gives the manuscript-level argument and results. The arXiv source package supports manuscript reproduction, and the ancillary benchmark archive contains the synthetic raw-shot data, subset diagnostics, disagreement tables, threshold maps, negative-control outputs, common-mode sensitivity maps, figures, scripts, manifests, and checksums. The scope is deliberately bounded. This is a synthetic/model-level benchmark and reproducibility package, not an experimental claim that Quantum Darwinism or Spectrum Broadcast Structure objectivity has been observed. Its purpose is to make explicit what future experimental raw-fragment evidence should provide before a plotted redundancy signature is treated as a claim of objective, independently accessible records.
This paper examines the role of Quantum Darwinism in explaining the emergence of classical objectivity within The Many-Worlds Interpretation of quantum mechanics. Decoherence explains why macroscopically distinct alternatives become practically inaccessible to local observers, but it does not by itself explain why multiple observers can independently agree on the same classical state. Quantum Darwinism strengthens this account by describing how information about stable pointer states becomes redundantly encoded in environmental fragments. However, this paper argues that redundancy is necessary for objectivity but not enough to explain objectivity. A simple qubit test model implemented in Qiskit is used to demonstrate redundant environmental coding. The discussion connects this claim to Strong Quantum Darwinism, Spectrum Broadcast Structure, and Quantum Discord, which sharpen the distinction between mere correlation and objective classical records. Finally, this paper proposes phase anti-redundancy as a future direction; While pointer state information becomes locally redundant, cross-branch phase information appears to remain globally encoded and increasingly difficult to recover. This suggests that branch objectivity may require not only redundant classical records, but also the fact that cross-branch phase information is encoded in global correlations rather than in locally accessible environmental fragments.
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.
Quantum Darwinism is a leading framework for explaining how classical reality emerges from a quantum universe—the environment redundantly encodes the pointer state information of quantum systems, enabling multiple independent observers to reach consensus about reality. However, a fundamental question has remained unanswered: Why does the environment possess this capacity for redundant recording? This paper proposes, within the information-potential conversion framework, that the redundant recording mechanism of quantum Darwinism is the physical realization of the information-potential conversion consensus protocol—the environment, acting as the physical medium for information-potential conversion, redundantly encodes system information, providing consistent conversion records for independent information processing units, from which classical reality emerges. Based on this interpretation, three testable quantitative predictions are proposed: (1) In superconducting quantum circuits, when system complexity exceeds a critical threshold, the efficiency of redundant recording should exhibit an abrupt transition—a signal of the information-potential conversion phase transition of the consensus protocol from “quantum critical” to “classically locked”; (2) There should exist a precise linear relationship between functional information and information-potential conversion energy consumption: each additional bit of functional information doubles the minimum heat dissipation required for record stabilization; (3) Consensus strength—the stability of classical reality—should be positively correlated with the number of information processing units participating in the consensus protocol, and a critical number exists at which the consensus phase transition occurs. quantum Darwinism, information-potential conversion, consensus protocol, classical emergence, redundant recording, functional information
We introduce an improved one-shot characterisation of randomness extraction against quantum side information (privacy amplification), strengthening known one-shot bounds and providing a unified derivation of the tightest known asymptotic constraints. Our main tool is a new class of smooth conditional entropies defined by lifting classical smooth divergences through measurements. A key role is played by the measured smooth Rényi relative entropy of order 2, which we show to admit an equivalent variational form: it can be understood as allowing for smoothing over not only states, but also non-positive Hermitian operators. Building on this, we establish a tightened leftover hash lemma, significantly improving over all known smooth min-entropy bounds on extractable randomness and recovering the sharpest classical achievability results. We extend these methods to decoupling, the coherent analogue of privacy amplification, obtaining a corresponding improved one-shot bound. Relaxing our smooth entropy bounds leads to one-shot achievability results in terms of measured Rényi divergences, tightening the bounds of [Dupuis, arXiv:2105.05342] and recovering state-of-the-art asymptotic i.i.d. error exponents. We show an approximate optimality of our results by giving a matching one-shot converse bound up to additive logarithmic terms. This yields an optimal second-order asymptotic expansion of privacy amplification under trace distance, establishing a significantly tighter one-shot achievability result than previously shown in [Shen et al., arXiv:2202.11590] and proving its optimality for all hash functions.