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.