Marginal information persistence is introduced as the condition that accumulated operational information under recursive composition approaches a finite, strictly positive asymptotic value only when the growth of available records exactly balances the loss of local distinguishability. In the canonical binary concatenated Information-Causality protocol, this criterion uniquely selects the Tsirelson bias η = 1/√2; for anisotropic coefficients, the corresponding critical set is ∑ᵢvᵢ² = 1. The exact Shannon recursion admits a positive series representation, while finite-depth corrections on the critical set are organized by the Rényi spectrum of the normalized squared coefficients, with collision entropy determining the leading correction rate. The central result concerns the physical pre-query register of the canonical singlet-based binary entanglement-assisted random access code (EARAC). For primitive bias v, the complete register satisfies the exact identity I(Aₖ:Rₖ) = G[(2v²)^(k/2)], whereas each target marginal satisfies I(Aⱼ:Rₖ) = G(vᵏ). At the critical value v = 1/√2, every individual target correlation vanishes exponentially with depth, while the mutual information between the complete database and the complete pre-query register remains exactly one bit at every depth. The difference between complete-register information and the sum of target marginals is resolved exactly as a sum of conditional mutual informations. The result therefore provides an exactly solvable example of marginal attenuation coexisting with depth-independent global information persistence in a concatenated quantum register. Two direct depth-two experiments on IBM ibm_marrakesh test this separation. A 128-circuit witness experiment yields the nonasymptotic certificate I(A₂:R₂) − ΣⱼI(Aⱼ:Yⱼ) ≥ 0.013245941 bit at 99.9% confidence, under explicitly stated repeated-shot, static-branch, and intended-observable assumptions. A second 864-circuit experiment performs full three-qubit tomography of the pre-query register. Physical maximum-likelihood reconstruction gives I(A₂:R₂) = 0.691632483 bit, ΣⱼI(Aⱼ:R₂) = 0.535829616 bit, and a reconstructed strong gap of 0.155802867 bit; an independent linear-inversion/PSD reconstruction agrees to within 4.06×10⁻⁴ bit. Bootstrap and matched-shot null simulations are reported only as estimator diagnostics; no strictly positive state-space confidence bound is claimed for the tomographic strong gap. The complete-register theorem motivates a quantum-information-processing perspective in which target-by-target readout can underestimate information retained by a composite quantum register. Connections to collective readout, information scrambling, quantum data hiding, and quantum error correction are discussed as related but distinct research directions. No computational speedup, fault-tolerance theorem, universal collective decoder, or characterization of the full quantum correlation set is claimed. This Zenodo record includes the manuscript together with the reproducibility package for the reported hardware analyses, including experimental data, reconstruction outputs, analysis scripts, and integrity checks.
Marginal information persistence is introduced as the condition that accumulated operational information under recursive composition approaches a finite, strictly positive asymptotic value only when the growth of available records exactly balances the loss of local distinguishability. In the canonical binary concatenated Information-Causality protocol, this criterion uniquely selects the Tsirelson bias η = 1/√2; for anisotropic coefficients, the corresponding critical set is ∑ᵢvᵢ² = 1. The exact Shannon recursion admits a positive series representation, while finite-depth corrections on the critical set are organized by the Rényi spectrum of the normalized squared coefficients, with collision entropy determining the leading correction rate. The central result concerns the physical pre-query register of the canonical singlet-based binary entanglement-assisted random access code (EARAC). For primitive bias v, the complete register satisfies the exact identity I(Aₖ:Rₖ) = G[(2v²)^(k/2)], whereas each target marginal satisfies I(Aⱼ:Rₖ) = G(vᵏ). At the critical value v = 1/√2, every individual target correlation vanishes exponentially with depth, while the mutual information between the complete database and the complete pre-query register remains exactly one bit at every depth. The difference between complete-register information and the sum of target marginals is resolved exactly as a sum of conditional mutual informations. The result therefore provides an exactly solvable example of marginal attenuation coexisting with depth-independent global information persistence in a concatenated quantum register. Two direct depth-two experiments on IBM ibm_marrakesh test this separation. A 128-circuit witness experiment yields the nonasymptotic certificate I(A₂:R₂) − ΣⱼI(Aⱼ:Yⱼ) ≥ 0.013245941 bit at 99.9% confidence, under explicitly stated repeated-shot, static-branch, and intended-observable assumptions. A second 864-circuit experiment performs full three-qubit tomography of the pre-query register. Physical maximum-likelihood reconstruction gives I(A₂:R₂) = 0.691632483 bit, ΣⱼI(Aⱼ:R₂) = 0.535829616 bit, and a reconstructed strong gap of 0.155802867 bit; an independent linear-inversion/PSD reconstruction agrees to within 4.06×10⁻⁴ bit. Bootstrap and matched-shot null simulations are reported only as estimator diagnostics; no strictly positive state-space confidence bound is claimed for the tomographic strong gap. The complete-register theorem motivates a quantum-information-processing perspective in which target-by-target readout can underestimate information retained by a composite quantum register. Connections to collective readout, information scrambling, quantum data hiding, and quantum error correction are discussed as related but distinct research directions. No computational speedup, fault-tolerance theorem, universal collective decoder, or characterization of the full quantum correlation set is claimed. This Zenodo record includes the manuscript together with the reproducibility package for the reported hardware analyses, including experimental data, reconstruction outputs, analysis scripts, and integrity checks.
This preprint develops marginal information persistence as a framework in which information itself is treated as the dynamical variable under repeated protocol composition. The central question is which information measures vanish, remain marginally finite, or persist exactly as concatenation depth increases. For the canonical binary concatenated entanglement-assisted random access code, the complete pre-query quantum register is characterized in closed form for arbitrary primitive correlation bias. At the Tsirelson point, Bob's complete register retains exactly one bit of Holevo information about the full database at every concatenation depth, while every individual target correlation decays exponentially. The work further distinguishes information stored in the quantum register from information operationally extractable from a single copy. For the associated critical Clifford ensemble, the accessible-information problem reduces exactly to a global unit-vector optimization. Explicit finite-depth bounds show that the optimal single-copy accessible information decays exponentially with concatenation depth, even though the complete-register Holevo information remains exactly one bit. The task-summed random-access score remains finite at the same critical point because it combines mutually exclusive query contexts. Complete-register Holevo information, target marginals, task-summed random-access information, and single-copy accessible information therefore occupy distinct asymptotic classes under the same recursion. The analysis connects marginal information persistence with Information Causality, Clifford observables, Holevo information, accessible information, entropy optimization, and earlier Fisher-information formulations of criticality in concatenated protocols. It also clarifies the relation to previous work on persistence, random-access codes, and information-theoretic stability. Depth-two consequences are tested experimentally on IBM Marrakesh and Quantum Inspire Tuna-17. Both platforms yield positive 99.9% finite-shot lower confidence bounds on an implemented decoder gap. Full-register physical tomography provides separate reconstruction estimates of the corresponding strong information gap. This Zenodo record contains the main manuscript and the Supplementary Information.
This version substantially revises and extends the March 2026 preprints on persistence under Information Causality. The manuscript reformulates the analysis around the exact information score of the canonical concatenated binary random-access-code protocol and develops an exact positive-series representation that leads to rigorous finite-depth bounds. These results establish a complete subcritical, critical, and supercritical classification of the concatenated Information-Causality dynamics. Correlations below the Tsirelson point exhibit asymptotic loss of accumulated information; correlations above it eventually violate Information Causality at finite depth; and the Tsirelson point is uniquely characterized by finite, nonzero asymptotic informational persistence. The new version also provides controlled finite-depth corrections at criticality, a rigorous treatment of the critical remainder, a mathematical branching generalization, a local Fisher-information interpretation of the quadratic scaling, and a universality result for smooth symmetric information-divergence measures. Numerical stability near the critical regime is treated explicitly. Compared with the March versions, the present manuscript therefore replaces the primarily asymptotic stability argument with a stronger exact finite-depth formulation and substantially clarifies the mathematical status and scope of the persistence interpretation. The scope remains deliberately restricted to the canonical isotropic concatenated Information-Causality/random-access-code setting. No claim is made that the construction independently characterizes the full quantum correlation set.
This paper introduces a foundational framework that reinterprets the position of quantum correlations within the space of possible physical theories. Rather than treating the Tsirelson bound (22) merely as a physical restriction, this work demonstrates that it emerges naturally as the unique global optimum for systems maximizing their persistence in environments of unbounded complexity. By integrating Information Causality (IC) with the Variational Principle of Persistence (VPP), we provide a unified information-geometric explanation for why quantum correlations occupy their precise position between classical and no-signaling theories. Key Contributions & Highlights: The Quantum Boundary as a Persistence Optimum: We prove that for task distributions with unbounded support, the maximum survival probability combined with informational benefit sharply isolates ρ=22 as the optimal state within the IC-constrained RAC framework. Exact Benefit–Metric Correspondence: We establish a direct identity linking the curvature of the informational benefit function to the classical Fisher information of the effective binary symmetric channel (G′′=IF/ln2). This reveals that IC is equivalent to the stability of the Fisher–Rao metric pullback under protocol concatenation, with the Tsirelson bound as the critical eigenvalue. Generalized Systemic Reduction Paradox (SRP): We identify a new optimization mechanism — a generalization of the SRP introduced in our prior VPP work — where the optimal state does not arise from classical diminishing returns (concave benefit vs. convex cost), but from a strictly convex benefit colliding with a hard viability wall, producing a corner optimum at the constraint boundary. Structural Necessity of Shannon Entropy: We demonstrate that Shannon entropy is the unique measure compatible with the concatenated RAC structure, grounded in its exclusive satisfaction of the strong chain rule. Alternatives such as Rényi entropy inherently fail to reproduce the correct quantum bounds. Significance: This manuscript bridges device-independent quantum foundations with information geometry and viability theory, offering a mathematical derivation of the Tsirelson bound as the optimal operational regime for compositional information processing under IC — within the CHSH scenario. Extensions to the full quantum correlation set remain an open question explicitly acknowledged in the paper. Reproducibility Package A reproducibility package accompanying this manuscript is available as a supplementary file. It includes a Jupyter/Colab notebook (ic_figures_colab_v3.ipynb) that reproduces Fig. 2 (Itotal vs. concatenation depth and viability horizon nmax vs. ρ) using high-precision arithmetic (mpmath, 80 decimal places) to avoid numerical cancellation near the Tsirelson point. The package also includes four CSV files containing the numerical data underlying Figs. 3–5. The notebook runs end-to-end on Google Colab with no manual configuration required.
We derive a causal law of observer-indexed entropy retrieval: retrieved reference information grows toward the supply available to a finite observer. A recorded admission-and-readout instrument produces orthogonal operational sectors from arbitrary coherent radiation; exact diary fidelity and decoupling give the flagged family, while branch errors bound its approximate realization. An exact quantum transfer channel yields the remaining-gap law and its continuous Lindblad evolution. The channel has an explicit unitary collision realization: admitted reference information moves from radiation to memory, while the capture record remains diary-blind. Its memory marginal is the Choi state of an erasure channel, giving an intrinsic positive-coherent-information boundary at retrieved fraction r = 1/2. A strip-holomorphic modular speed limit bounds how rapidly the channel can activate, with the hyperbolic tangent selected uniquely at extremality. The integrated retrieval dose is invariant under a change of clock, making the Kerr lapse cancellation one geometric instance of a general result. At fixed supply and kinetic rate under an affine clock, the transfer and activation laws compose into a closed proper-time retrieval trajectory on the equality branch, identify that trajectory as the sharp fastest-retrieval envelope, and fix the earliest admissible retrieval horizon. In the continuum formulation, Araki relative entropy defines retrieval on restricted observable algebras, and a split Type-I factor provides the regulated density-operator representation. The experimental program follows the same causal sequence through retrospective analysis and staged intervention. Tier 0A determines whether archived analog-horizon data retain the shot-level, timing, protocol, and processing information needed to ask the retrieval question. Tier 0B then tests whether qualifying data contain observer-, protocol-, readout-, or analysis-operator-indexed correlation structure that standard pooling or filtering suppresses. A positive Tier 0B result identifies retrieval-relevant structure and motivates Tier 1; it does not by itself confirm the complete law. The prospective Tier 1 BEC program targets the held-out g²(t₁,t₂) surface, bandwidth scaling, and protocol dependence. Independent activation readout tests the modular bound and tanh rigidity. Reference-tagged supply and memory test cumulative dose, composite-hazard recovery, degraded-channel ordering, and two-memory allocation. The v2 verification artifact provides synthetic benchmarks and numerical consistency checks at bond dimensions D = 4 and D = 8 for the transfer law, inverse recovery, finite-resolution robustness, and matched adversarial alternatives. Standard RT/HRT geometry and a separate accessibility functional place the retrieval dynamics within holographic language. The retrieval law closes the observer-local path between global encoding and retained access. Information conservation, radiation encoding, causal collection, finite-observer retrieval, formal reconstruction, and comprehension are distinct physical operations. The black-hole information problem contains a proper-time access dynamics that global entropy accounting alone does not determine. Physical complexity enters through encoding, supply, clock, activation, transfer, retention, and observation. Those layers locate each source of retrieval failure instead of idealizing it away. The theory orders the encoding, supply, and retrieval clocks; distinguishes late supply from early supply through their overlap with future transfer dose; and gives an exact finite-lifetime retrieval criterion. Here rᵢ(τ) is the normalized reference information retained by observer i, eᵢ(τ) is the causally available supply, kᵢ(τ) is the composite access hazard, and τ_char is the characteristic activation scale.
This work establishes the Black-Hole Information Sector Connection of Return Theory (RT) in finite-dimensional quantum hosts. It applies RT’s typed observational roles—observer, host stage, validation reference, representation reference, access, record, and reconstruction—to black-hole information problems while retaining the supplied quantum dynamics. The central result is a three-level conditional classifier that locates an apparent conflict at its first failed level: host-state feasibility, same-run record and access compatibility, or equivalence between reconstructed images. In the Hayden–Preskill host, an old black hole is maximally entangled with early radiation, a one-qubit diary is scrambled by a fixed Haar unitary, and recovery is evaluated through two access channels on the same dynamics: emitted radiation R alone and the combined system ER. Classical diary content is assessed by exact binary minimax error, while coherent recovery is assessed separately by Petz entanglement fidelity. Across 36 fixed unitaries with total coherent register sizes of 8, 10, and 12 qubits, ER reaches both declared recovery thresholds after two emitted qubits in every run, whereas R reaches them only after complete emission. In the selected 12-qubit realization at k = 2, adding access to E changes the minimax error from 0.472087 to 0.031872 and the Petz fidelity from 0.252652 to 0.948306, with the unitary dynamics held fixed. The AMPS audit is evaluated under two explicit factorization declarations. With independent tensor factors E, b, and b̃, simultaneous exact Bell targets for Eb and bb̃ define an empty marginal-feasibility set. The conflict is therefore classified at the host-state level. When b̃ is instead declared as a logical algebra encoded in E by a [[4,1,2]] quantum error-correcting code, one consistent joint state exists and the issue moves to access, records, and reconstruction. Within the encoded realization, the disjoint logical-Z representatives Z₁Z₃ and Z₂Z₄ generate exact, order-independent records against a stable external validation reference and satisfy the declared RT operational-objectivity conditions on the binary logical-codeword ensemble. These objective classical records reduce logical coherence, while persistent coherent extraction transfers full logical fidelity to the first output and leaves only Bell fidelity 1/4 for a later E-only output. The calculation therefore distinguishes redundant objective records of commuting classical content from reconstruction of the complete logical quantum state. The finite connection further includes five modules: a sequential evaporation host that instantiates the RT Time role and distinguishes monotone admitted-record histories from observer accesses that lose records; an exact independent-factor Bell-fidelity frontier with symmetric optimum 3/4; a graded early-radiation-entanglement sweep that moves the ER and R recovery thresholds in opposite directions; a three-qutrit code whose two-share recoverability coexists with record redundancy one; and typing witnesses showing how verdicts change when minimax admission, joint realization, records, occupied order, or certified validation are replaced by inequivalent proxies. The resulting sector connection shows how factorization, observer-supported access, reference stability, record admission, and reconstruction jointly determine the consistent observational reading of a supplied black-hole information model. It provides formal statements, executable finite-dimensional witnesses, independent numerical verification, machine-readable results, complete source code, and cryptographic checksums.
Extracting operationally meaningful quantities such as von Neumann entropy and mutual information is central to characterizing many-body quantum systems, yet experiments and numerics often provide direct access only to integer Rényi entropies. Whereas conventional extrapolation relies on a prescribed fitting ansatz, in PRL 137, 100202 we introduced an alternative approach based on stabilized analytic continuation (SAC), which avoids such an ansatz and is inherently robust to noise. Here, we further develop this framework in the noiseless setting and benchmark its performance across several nontrivial many-body problems. We first show that, besides the intrinsic ambiguity of reconstructing the von Neumann entropy from finitely many Rényi samples, analytic continuation can also fail because of genuine nonanalyticities in the Rényi function arising from zeros of Trρz. We derive universal zero-free domains for Trρz and systematically sharpen these bounds using additional spectral information about ρ. We then benchmark the performance of SAC in three settings: (i) extracting topological entanglement entropy from DMRG Rényi data for the toric code and Kagome Heisenberg models; (ii) reconstructing the mutual information between disjoint intervals in a (1+1)d compact-boson CFT; and (iii) recovering finite-time ballistic growth of the von Neumann entropy from integer Rényi entropies that cross over toward subballistic growth in diffusive quantum dynamics. In the latter setting, analytic continuation is expected to fail in the long time limit. We illustrate this mechanism with a physically motivated two-sector model, where the asymptotic separation between von Neumann and higher-Rényi growth is accompanied by a zero of Trρz pinching the real axis at z=1, thereby producing a first-order Rényi phase transition.