Redundancy-Induced Nonlocality of Frame Information in Branching Decoherence Dynamics Exact Leakage Curves, a Universal Record-to-Frame Exclusion Theorem, and a Secret-Sharing Interpretation
This paper quantifies a structural exclusion principle in quantum decoherence dynamics: when environment fragments carry redundant classical records of a pointer observable, information about any conjugate observable (the frame variable) is forced into nonlocal, high-order correlations. The work fills a gap in the quantum Darwinism program (Zurek 2003, 2009), which has thoroughly characterized how pointer information is redundantly broadcast to environment fragments but has not provided a quantitative treatment—with explicit constants and tight bounds—of the complementary question: what happens to conjugate (phase/coherence) information as pointer redundancy increases. The central result is an exact complementarity law across every bipartition, demonstrating that pointer accessibility in the unobserved complement is precisely the resource that suppresses frame accessibility to a verifier. This identity is fully consistent with the no-hiding theorem (Braunstein and Pati 2007), which guarantees that frame information remains in principle recoverable from the environment; the contribution here is to quantify how the required coalition size scales with pointer redundancy according to an explicit exponential curve, establishing the resource cost of recovery that the no-hiding theorem leaves open. Key results include a quantitative redundancy–leakage tradeoff with explicit constants, a universal record-to-conjugate exclusion theorem valid for general pointer-preserving Stinespring isometries without product-branching or conditional-independence assumptions, and a formal cross-domain equivalence establishing that physical branching dynamics induce a ramp secret-sharing and data-hiding access structure for the frame bit. The secret-sharing structure connects to the quantum data hiding literature (DiVincenzo, Leung, and Terhal 2002; Eggeling and Werner 2002; Lami, Palazuelos, and Winter 2018), with the distinction that the hiding here is not artificially constructed but arises naturally from decoherence with redundant pointer records, parameterized by a single physically measurable quantity—the decoherence factor of the unobserved complement. The effective dephasing channel induced by fragment erasure has a structural analogy to the erasure channels studied in fault-tolerant quantum error correction, with pointer redundancy playing the role of an effective code distance for conjugate observables; the key difference from the QEC threshold setting is that the encoding is imposed by nature rather than designed, and the access structure may not be known a priori. The paper also complements information-theoretic equilibration approaches (Short and Farrelly 2012) and discord-based irreversibility arguments (Zurek 2018) by providing explicit coalition-size thresholds tied to physical parameters rather than generic typicality bounds or discord measures. The work formalizes the transition from quantum coherence to classical objectivity by elevating complementary information bounds to exact, model-independent structural identities, while acknowledging that the branching-family starting point inherits the system–environment partition assumed by the quantum Darwinism framework.
This paper quantifies a structural exclusion principle in quantum decoherence dynamics: when environment fragments carry redundant classical records of a pointer observable, information about any conjugate observable (the frame variable) is forced into nonlocal, high-order correlations. The work fills a gap in the quantum Darwinism program (Zurek 2003, 2009), which has thoroughly characterized how pointer information is redundantly broadcast to environment fragments but has not provided a quantitative treatment—with explicit constants and tight bounds—of the complementary question: what happens to conjugate (phase/coherence) information as pointer redundancy increases. The central result is an exact complementarity law across every bipartition, demonstrating that pointer accessibility in the unobserved complement is precisely the resource that suppresses frame accessibility to a verifier. This identity is fully consistent with the no-hiding theorem (Braunstein and Pati 2007), which guarantees that frame information remains in principle recoverable from the environment; the contribution here is to quantify how the required coalition size scales with pointer redundancy according to an explicit exponential curve, establishing the resource cost of recovery that the no-hiding theorem leaves open. Key results include a quantitative redundancy–leakage tradeoff with explicit constants, a universal record-to-conjugate exclusion theorem valid for general pointer-preserving Stinespring isometries without product-branching or conditional-independence assumptions, and a formal cross-domain equivalence establishing that physical branching dynamics induce a ramp secret-sharing and data-hiding access structure for the frame bit. The secret-sharing structure connects to the quantum data hiding literature (DiVincenzo, Leung, and Terhal 2002; Eggeling and Werner 2002; Lami, Palazuelos, and Winter 2018), with the distinction that the hiding here is not artificially constructed but arises naturally from decoherence with redundant pointer records, parameterized by a single physically measurable quantity—the decoherence factor of the unobserved complement. The effective dephasing channel induced by fragment erasure has a structural analogy to the erasure channels studied in fault-tolerant quantum error correction, with pointer redundancy playing the role of an effective code distance for conjugate observables; the key difference from the QEC threshold setting is that the encoding is imposed by nature rather than designed, and the access structure may not be known a priori. The paper also complements information-theoretic equilibration approaches (Short and Farrelly 2012) and discord-based irreversibility arguments (Zurek 2018) by providing explicit coalition-size thresholds tied to physical parameters rather than generic typicality bounds or discord measures. The work formalizes the transition from quantum coherence to classical objectivity by elevating complementary information bounds to exact, model-independent structural identities, while acknowledging that the branching-family starting point inherits the system–environment partition assumed by the quantum Darwinism framework.
We prove that decoherence is practically irreversible through a purely information-theoretic mechanism that invokes neither the second law of thermodynamics nor Hilbert space dimensionality arguments. In branching decoherence dynamics, the frame variable (the relative phase between pointer sectors) is stored as a ramp secret in an access structure induced by redundant environment records. A bounded-memory reversal agent faces a scouting–recovery complementarity: measuring fragments to identify records induces back-action that degrades subsequent frame recovery. This tradeoff yields a meta-Nyquist gap governing the agent's net coherent-information gain rate. When pointer redundancy exceeds a critical threshold, the gap turns negative and the agent enters a renewal trap — an absorbing regime in which coherent qubits are lost faster than useful fragments can be incorporated. The main result (the Informational Irreversibility Theorem) shows that frame recovery probability is exponentially close to chance for any adaptive LOCC protocol. Five extensions establish universality: channel independence via Lindbladian spectral gaps, robustness under partial quantum error correction, a passive–active duality between the meta-Nyquist threshold and the fault-tolerance threshold, an optimal redundancy analysis revealing that quantum Darwinism maximizes the informational cost of reversing classicality, and a thermodynamic separation via an explicit pure-state model with zero entropy. Additional results include a categorical duality functor, tight spectral bounds for physical noise channels, multi-party verifier networks, and an adversarial extension connecting to Byzantine fault tolerance.
Quantum Darwinism explains classical objectivity through the redundant spread of information across fragments of the environment. This note studies the opposite regime: an environment whose record is distributed through entangled correlations rather than independent fragments. The model considers a mediator carrying a which-branch displacement record while being purified by an unobserved environment. The central question is: how much of the original quantum information can an observer recover when only part of the environment is accessible? The paper proves that any fragment excluding the record-bearing mode contains exactly zero which-branch information. No individual fragment carries a readable record, and redundancy disappears completely. A closed-form partial-information curve is derived. The amount of accessible evidence depends only on the captured fraction of the environment purification and is governed by a single rational function. The curve connects two limits: the minimal information available to an external observer and the full information recovered when the complete environment is captured. A precise threshold is obtained: strong environmental entanglement requires capturing almost the entire environment before objectivity emerges. The results show that decoherence and objectivity are not equivalent. The same loss of coherence can either create a public, redundantly observable fact or leave information hidden entirely in nonlocal correlations. The structure of the environment determines whether quantum information becomes an objective classical fact or remains inaccessible. All results are derived in exactly solvable Gaussian models, verified analytically and numerically, and accompanied by a reproducibility package.
This paper is the seventh in a sequence developing an interpretive framework that redescribes physical reality as a causally consistent history of information updates rather than as a collection of objects independently existing within spacetime. Two prior results set up its question. The third paper (Uchida, 2026c) characterized public finality—recoverability of a committed fact by many mutually independent accessors, non-disturbingly and repeatably—as a broadcast redundancy: an effectively classical pointer value proliferated across many disjoint environmental fragments, approximately in the form of spectrum broadcast structure. The fourth paper (Uchida, 2026d) read the evaporating black hole as the regime in which a committed interior record crosses, at the Page time, from hidden to recoverable encoding, reconstructible from the radiation through entanglement-wedge reconstruction; and the parent framework (Uchida, 2026a, §10) had flagged holographic quantum error correction as a place its language of protected, redundantly encoded, locally recoverable records might find content. The present paper makes that content precise as a structure theory of recoverable records, kept separate from any claim of new physics or any suggestion that the framework derives the underlying results. The central claim is conditional and structural. The framework does not claim that every recoverable piece of information is a code, that the universe is literally a quantum error-correcting code, or that AdS/CFT describes our universe. It claims that when information qualifies as an erasure-recoverable record because an encoding preserves a logical degree of freedom—or a logical operator algebra—against a specified family of physical erasures, then that recoverability is formalized as an exact or approximate quantum error-correcting structure, in which each correctable erasure leaves an authorized complementary region supporting a recovery map. That formalization is itself an established mathematical classification; the paper's interpretive contribution is to read the series' recoverable record through it, and to set it against a second, distinct architecture. Broadcast redundancy makes the same effectively classical value legible from many disjoint fragments; erasure-correcting redundancy protects noncommuting logical quantum information by keeping each correctable erased region ignorant of—approximately private with respect to—the protected logical content, while suitable remaining regions recover it. The formal engine of this distinction is the duality between correctability and privacy (the information–disturbance tradeoff), not the no-cloning theorem; no-cloning, with the access structure of quantum secret sharing, survives as the reason independently usable recovery regions for arbitrary unknown logical quantum information must overlap. Three commitments keep the reading honest. First, it separates three senses of "record"—protected logical quantum information, a committed classical record, and a publicly final record—and insists that recovery or decoding restores access to logical quantum information but does not, by itself, commit a classical value; commitment additionally requires measurement, pointer imprinting, decoherence, and stabilization (Uchida, 2026f), and public finality a still later broadcast. Second, holographic quantum error correction is offered as the sharpest worked model of the distinction, with exactly solvable tensor-network toy codes (exact within the model) kept distinct from realistic semiclassical holographic reconstruction beyond those limits (generally treated as approximate), and both from the further question of state-dependence. Third, code-recoverability does not by itself imply public finality: protected quantum information can be recoverable from authorized regions while remaining non-public. The framework relabels code structure as the architecture of recoverability and derives none of it, making no independent empirical prediction. Its disciplined thesis is that codes protect recovery; broadcasts create publicity—different architectures that a single physical history may nonetheless realize in sequence.
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.