This record contains aexpanded research-program manuscript and reproducible companion package for a finite-dimensional framework connecting recoverability frustration, entropy-loss calculus, observer screening, path consistency, higher-order memory diagnostics, and quantum statistical comparison in quantum channel networks. The work separates microscopic recoverability from observer-visible distinguishability. The formal core includes a four-level ladder between visible defects, microscopic recovery defects, entropic proxies, and entropy-loss actions; structural tests for post-processing monotonicity, hidden convex channel mixing, tensor-product factorization, continuity, path consistency, and observer monotonicity; and a probabilistic observer layer based on Blackwell garbling, forward Le Cam deficiency, visible Kullback-Leibler rates, and visible Fisher information. The accompanying code regenerates figures, validation summaries, and theorem-level regression tests.
This record contains aexpanded research-program manuscript and reproducible companion package for a finite-dimensional framework connecting recoverability frustration, entropy-loss calculus, observer screening, path consistency, higher-order memory diagnostics, and quantum statistical comparison in quantum channel networks. The work separates microscopic recoverability from observer-visible distinguishability. The formal core includes a four-level ladder between visible defects, microscopic recovery defects, entropic proxies, and entropy-loss actions; structural tests for post-processing monotonicity, hidden convex channel mixing, tensor-product factorization, continuity, path consistency, and observer monotonicity; and a probabilistic observer layer based on Blackwell garbling, forward Le Cam deficiency, visible Kullback-Leibler rates, and visible Fisher information. The accompanying code regenerates figures, validation summaries, and theorem-level regression tests.
This paper develops an engineering-facing diagnostic bridge between the recoverability-based theory of structural irreversibility and practical evaluation in quantum error correction and monitored quantum platforms. The central problem addressed in this work is that structural recoverability, threshold-like margins, false-collapse, and Recoverability Index–type diagnostics provide useful theoretical distinctions, but they do not by themselves specify how experimentalists or engineers should organize platform-level data, compare regimes, or report recoverability-relevant behavior. The paper therefore introduces a proxy map from structural quantities to engineering diagnostics. Generalized recoverability is associated with recovery performance, decoder-assisted restoration, logical-state fidelity, and related reconstruction-sensitive quantities. Threshold-like margins are associated with entropy-like, noise-sensitive, or transition-localizing proxies. False-collapse is associated with mismatch windows in which collapse-like or transition-like indicators become strong while recovery-sensitive records still indicate residual recoverability. The paper also proposes a minimal diagnostic architecture for QEC and monitored platforms. This architecture consists of four layers: an appearance-sensitive proxy, a recoverability-sensitive proxy, a threshold-like locator, and a window-based reporting layer. Together, these layers support regime classification, early-warning detection, false-collapse-like mismatch identification, finite-window reporting, and cross-regime comparison. A QEC-oriented realization is presented to show how syndrome histories, logical-failure markers, logical recovery performance, entropy-like proxies, false-collapse window frequencies, and RI-like summaries can be organized into a coherent diagnostic workflow. The goal is not to replace detailed decoder analysis or hardware-specific benchmarking, but to provide a bridge-level language for deciding what should be logged, estimated, compared, and reported when studying the approach to effective irreversibility in error-managed quantum systems. This work should be read as an engineering and experimental companion to the broader recoverability program. Its contribution is to make structural irreversibility operationally legible by mapping theoretical quantities to measurable, computable, and reportable diagnostics in contemporary quantum platforms.
This repository contains the source files for the paper: Irreversibility and Ordering in Classical Observability of Quantum Structure The paper develops a formal framework explaining when and why classical statistics extracted from noisy quantum systems retain architecture-dependent information, and when they inevitably lose it. Building on a trilogy of earlier empirical and operational studies, the work introduces the diagnostic pipeline as a fundamental object of analysis: a fixed composition of a noise channel, a measurement choice, and a deterministic classical feature map. Such pipelines are shown to form a preorder under information-degrading transformations, induced by standard monotonicity principles. Within this framework, several phenomena observed empirically are shown to be structural necessities rather than numerical accidents: Feature-dependent collapse arises from deterministic feature post-processing below the identity (full-distribution) pipeline. Basis-dependent diagnostic blindness corresponds to incomparability between pipelines using different fixed measurement bases. Irreversible loss of distinguishability follows from genuine noise degradation and cannot be undone by downstream measurement or feature choices. The paper proves general monotonicity results, establishes generic incomparability of measurement bases, and derives falsifiable predictions linking noise–basis alignment to collapse thresholds for population-encoded distinctions. Minimal empirical consistency checks against the companion trilogy datasets are provided. The framework yields practical design principles for lightweight diagnostics in NISQ-era experiments, clarifying fundamental limits on classical access to quantum structure under realistic constraints. Contents Irreversibility_and_Ordering_in_Classical_Observability_of_Quantum_Structure.pdfFinal compiled manuscript. Irreversibility_and_Ordering_in_Classical_Observability_of_Quantum_Structure.texLaTeX source (self-contained; bibliography and figures embedded). Relation to Other Work This paper provides the formal, structural foundation for a companion trilogy on architecture-sensitive classical fingerprints under noise, including: Feature-dependent collapse under fixed pipelines Basis-dependent diagnostic blindness Selective transcription as an operational principle The present work unifies and explains these results within a single information-theoretic preorder framework. Status Submission-ready manuscript.Intended for submission to a peer-reviewed journal in quantum information theory. Licence See LICENSE.
This preprint studies operational equivalence and recovery in a twenty-dimensional residual system with a fixed four-sector representation of dimensions 8, 8, 2 and 2. It distinguishes what a declared observation algebra can detect, what a channel preserves, and what a retained classical record makes recoverable. Record-resolved criteria separate useful reversible-error records from unrelated labels and measured-sector records, even when the discarded channel is the same. For a specified coarse-record model, the work identifies exactly recoverable observable algebras and proves an optimal entanglement-fidelity expression on a fixed four-state orbit code. Preparation-aware recovery is then classified over all one-use isometric subspaces in the stated family. An explicit ten-dimensional marker code attains exact recovery under a specified stochastic record where the whole-sector support restriction permits only eight dimensions. Its encoding, nonbaseline marker measurement, and conditional inverse operations remain supplied resources. Scoped Lean evidence verifies the explicit arbitrary-complex-matrix recovery construction and related finite identities; it does not formalize the general complete-positivity/reference framework or optimal-dimension theorem. The recorded-error model is not derived from the companion exchange-timing model, and no protection of its long pulse sequence is claimed.
Recoverability of quantum coherence is formulated as a family of success-constrained operational frontiers indexed by a state, channel, or multi-time process, a task, an interface, and an exact- or minimum-success convention. The framework separates state ceilings from experimentally reachable performance and distinguishes restrictions due to control, time, uncertainty, environmental records, memory, and inaccessible channel structure. The manuscript develops a typed simulation preorder on complete response families, a compiler-relative closure order, and a directed response deficiency on success-tagged behaviors. It gives exact static trace-norm ceilings, success-frontier geometry, explicit memory and record benchmarks, primal-dual finite-experiment separation programs, and reference-assisted channel-lineage witnesses. “Shadow activation” denotes the interface-relative exposure of a pre-existing process distinction under an enriched tester interface; it is not proposed as a new microscopic law. Version 1.0 follows a claim-by-claim bibliography and content audit. It is a separate work, conceptually derived in part from the earlier Italian criterion DOI 10.5281/zenodo.21333992, but it is neither a translation nor a new version of that record. This is a theoretical preprint and has not been peer reviewed.
How do you decide which quantum channel keeps more information? There is more than one way to ask. You can compare how well two states remain distinguishable, how much information an ensemble retains, or how sensitively an output responds to a small change. These are different mathematical questions. This paper asks when they give the same ordering of quantum channels. The comparison is complete: an input may share correlations with a separate, finite quantum reference, and the reference can be as large as needed. That extra system makes the question richer than looking at the channel output alone. For finite-dimensional channels, the manuscript claims an equivalence among complete comparisons expressed through relative entropy, Holevo information, BKM and SLD Fisher information, and measured chi-squared divergence. It also studies the complementary channel—the part of a Stinespring description that goes to the environment—and claims that the information order reverses under genuine complementation. The second part asks whether these comparisons can be decided by an algorithm. The paper claims coRE-completeness for a class of strictly positive rational-complex measurements, and that no computable bound on reference size suffices for the stated SLD comparison. It also claims fixed-additive-accuracy inapproximability for a bounded entropy defect. These are claims about precisely specified mathematical decision problems, not a claim that every practical channel comparison is impossible. The companion paper, From complete channel order to capacity and one-way distillation, explores consequences for communication and distillation using the order results developed here. See the manuscript page for a further introduction and citation details.
When quantum information initially stored in a local qubit disappears, it need not be lost: it may have moved into nearby degrees of freedom or become inaccessible to shallow local control. We introduce finite-window recoverability as an operational channel benchmark that separates these possibilities. It compares optimal recovery from the target site, recovery by a bounded-depth decoder on a finite window, and the unrestricted optimum for that window. Its operational component, local variational recovery, uses local state preparation, window-local control, and target-qubit Pauli readout to certify recoverable memory beyond the target and quantify how much of the same-window advantage is accessible to shallow control. In a disordered kicked-Ising Floquet chain, a depth-6 decoder on a five-site window realizes Q0opt<Q2shallow<Q2opt across the crossover regime, with positive certified gain for most disorder realizations and substantial shallow-accessibility fractions. The signal differs from target-site persistence and reconstructed coherent-information increments. Positive radius-2 gain also persists when the task is embedded in longer open chains using an independent tensor-network backend. Guided by this hierarchy, we test a carrier-deletion task in which the original target register is reset after the dynamics. A depth-8 decoder repairs the input from a radius-3 surrounding halo with held-out median Favg=0.758, above the single-qubit classical benchmark 2/3, and outperforms optimal one-, two-, and three-site halo-subwindow counterfactuals. These results establish finite-window recovery as a local-control benchmark for off-site quantum memory, diagnosing both where local quantum information remains and whether bounded-depth control can refocus it.