Within flagged unitary–replacer channels, optimized recovered entanglement fidelity is spectrally incomplete: it fixes weighted transparency but not the hidden replacer spectrum or conditional Choi entropy. For a resolved positive-probability sector with known transparency 0 ≤ a < 1, we prove a family-specific noncommutative triangular transfer from conditional Choi moments M2, . . . , Md to hidden-spectrum power sums. Recursive inversion, Newton–Girard identities, and a rank-one step recover the unordered replacer and conditional Choi spectra, including multiplicities and zero eigenvalues. The order-d threshold is sharp within the consecutive sector-resolved hierarchy: every truncation below d fails local injectivity on positive simple-spectrum chambers. At fixed dimension and flag distribution, we also determine the complete homogeneous and heterogeneous fixed-fidelity Choi-entropy intervals, including lower/equality structure, interval filling, and heterogeneous optimizer geometry, while retaining the established quantum-Fano upper envelope. With transparency unknown, the consecutive map (a, λ) ↦ (M2, . . . , Md+1) is generically locally invertible on the positive ordered simple-spectrum interior for every d ≥ 2. Global uniqueness holds in d = 2, 3; for the specific map F4 = (M2, M3, M4, M5), d = 4 exhibits an exact open two-sheet ambiguity with exactly two ordered physical inverse branches, while d ≥ 5 remains unresolved. At a = 1, the moments lose all replacer-spectrum information. The results are family-specific and assume sector-resolved access; they do not imply universal channel reconstruction, tomography, or resource optimality.
Within flagged unitary–replacer channels, optimized recovered entanglement fidelity is spectrally incomplete: it fixes weighted transparency but not the hidden replacer spectrum or conditional Choi entropy. For a resolved positive-probability sector with known transparency 0 ≤ a < 1, we prove a family-specific noncommutative triangular transfer from conditional Choi moments M2, . . . , Md to hidden-spectrum power sums. Recursive inversion, Newton–Girard identities, and a rank-one step recover the unordered replacer and conditional Choi spectra, including multiplicities and zero eigenvalues. The order-d threshold is sharp within the consecutive sector-resolved hierarchy: every truncation below d fails local injectivity on positive simple-spectrum chambers. At fixed dimension and flag distribution, we also determine the complete homogeneous and heterogeneous fixed-fidelity Choi-entropy intervals, including lower/equality structure, interval filling, and heterogeneous optimizer geometry, while retaining the established quantum-Fano upper envelope. With transparency unknown, the consecutive map (a, λ) ↦ (M2, . . . , Md+1) is generically locally invertible on the positive ordered simple-spectrum interior for every d ≥ 2. Global uniqueness holds in d = 2, 3; for the specific map F4 = (M2, M3, M4, M5), d = 4 exhibits an exact open two-sheet ambiguity with exactly two ordered physical inverse branches, while d ≥ 5 remains unresolved. At a = 1, the moments lose all replacer-spectrum information. The results are family-specific and assume sector-resolved access; they do not imply universal channel reconstruction, tomography, or resource optimality.
Within flagged unitary–replacer channels, optimized recovered entanglement fidelity is spectrally incomplete: it fixes weighted transparency but not the hidden replacer spectrum or conditional Choi entropy. For a resolved positive-probability sector with known transparency 0 ≤ a < 1, we prove a family-specific noncommutative triangular transfer from conditional Choi moments M₂,...,M_d to hidden-spectrum power sums. Recursive inversion, Newton–Girard identities, and a rank-one step recover the unordered replacer and conditional Choi spectra, including multiplicities and zero eigenvalues. The order-d threshold is sharp within the consecutive sector-resolved hierarchy: every truncation below d fails local injectivity on positive simple-spectrum chambers. At fixed dimension and flag distribution, we also determine the complete homogeneous and heterogeneous fixed-fidelity Choi-entropy intervals, including lower/equality structure, interval filling, and heterogeneous optimizer geometry, while retaining the established quantum-Fano upper envelope. With transparency unknown, the consecutive map (a, λ) ↦ (M₂,...,M_{d+1}) is generically locally invertible on the positive ordered simple-spectrum interior for every d ≥ 2. Global uniqueness holds in d = 2, 3; for the specific map F₄ = (M₂, M₃, M₄, M₅), d = 4 exhibits an exact open two-sheet ambiguity with exactly two ordered physical inverse branches, while d ≥ 5 remains unresolved. At a = 1, the moments lose all replacer-spectrum information. The results are family-specific and assume sector-resolved access; they do not imply universal channel reconstruction, tomography, or resource optimality.
A finite-dimensional physical processor cannot exactly program a continuous family of distinct unitary channels. We show that this obstruction becomes quantitative when the target channel is stored in a normalized Choi state and its output observables are reconstructed by sampling physical channels and classically post-processing their measurement outcomes. For arbitrary d-dimensional channels, we construct a target-independent exact reconstruction protocol and prove the optimal one-copy sampling overhead, which grows quadratically with system dimension. We further prove the sharp fixed-d law that the excess overhead vanishes inversely with the number of identical Choi programs. The upper bound combines deterministic port-based teleportation with a quasi-decomposition that corrects its depolarizing distortion. The converse maps any low-overhead reconstruction protocol to a physical learner of unknown unitaries and uses local quantum estimation to recover the same leading coefficient. These results recast the universal no-programming obstruction as a quantitative trade-off between quantum program memory and classical sampling, with a leading cost that reflects the locally learnable unitary degrees of freedom.
A quantum instrument is not determined by its induced positive-operator-valued measure (POVM) and nonselective channel, yet the global control consequences of this residual branch structure remain incompletely characterized. We study the nondegenerate fiber of instruments with both marginals fixed. The subgroup of outcome permutations preserving the effect tuple acts as the deck group of the ordered outcome-frame covering, and closed loops define a conjugacy-class-valued monodromy representation. Nontrivial monodromy obstructs continuous global branch labeling based only on local process continuity. We construct an explicit cyclic family with a uniform m-outcome POVM and a fixed completely dephasing channel, derive robust overlap-based reconstruction and a conservative finite-sample guarantee, and certify entangled conditional Choi states although the nonselective channel is entanglement breaking. For a noisy cyclic family, we prove an exact minimax separation in conditional feedback: recovery rules restricted to the two marginals and the reported local outcome have optimal worst-sector entanglement fidelity 1/m, whereas an outcome-frame-aware controller attains 1-lambda+lambda/m. The exact minimax value of the missing outcome-frame information is therefore (1-lambda)(1-1/m). The results remain entirely within standard finite-dimensional quantum mechanics and are positioned as an instrument-level, permutation-valued specialization of established Kraus-frame/channel holonomy with a distinct global-labeling and feedback limitation.
Inverse Geometry and Identifiability of Protocol Ratios in Dynamical-Decoupling Spectroscopy This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Building on reduced observation geometry, it asks the inverse question: given an observed protocol ratio, what spectral information is uniquely reconstructible, what is only weakly constrained, and what is fundamentally unrecoverable? The analysis is formulated in terms of the Jacobian of the reduced observation map and the associated Fisher information matrix. The paper shows that the inverse problem separates into three qualitatively distinct sectors. In the regular sector, the ratio map is locally invertible and reconstruction accuracy is controlled by a Cramér–Rao bound. For the pure power-law family S(ω)=Aω^α, the ratio R(α)=2/(1+2^{-α}) is strictly monotone, admits a closed-form inverse, and yields an explicit conditioning profile through the slope ∂_αR. In the degenerate sector, exemplified by the white-noise family S(ω)=A, the protocol ratio is exactly pinned to R=1 for all amplitudes. The Fisher information for the amplitude direction vanishes identically, so amplitude reconstruction from ratio data alone is impossible. This gives an exact no-go theorem for identifiability from protocol ratios in the white-noise case. In the branched sector, non-injectivity of the inverse map produces intrinsic ambiguity. Using the mixed family S(ω; α1, α2, w)=wAω^{α1}+(1−w)Aω^{α2}, the paper shows both global non-injectivity and local fold branching. Along an explicit symmetric slice, the ratio develops a fold at Rc=4/3, so nearby observed values admit two distinct inverse branches. These results establish protocol ratios as inverse-geometric probes with sharply defined domains of identifiability and intrinsic limits of spectral reconstruction. The framework clarifies the difference between unique inference, exact information collapse, and multi-valued inversion, and provides a basis for future extensions to multi-ratio, mixed-spectrum, and frequency-resolved inverse design. V2:This work develops an inverse identifiability framework for protocol ratios in dynamical-decoupling spectroscopy. Interpreting the reduced observation map through Fisher information, we show that the inverse problem separates into three sectors: regular (locally invertible with Cramér–Rao bounds), degenerate (exact collapse with vanishing Fisher information), and branched (non-injective with multiple solutions). Using power-law and mixed spectral families, we derive explicit inversion formulas, a white-noise no-go theorem, and fold-induced branching. The framework establishes protocol ratios as inverse-geometric probes with sharply defined limits of spectral reconstruction.
Reduced descriptions of open quantum systems are intrinsically non-identifying: many different bath Hamiltonians, initial bath states, dilations, and microscopic couplings may generate the same reduced channel on the accessible system. This paper introduces a finite-dimensional certification protocol for this non-identifiability using the normalized Choi response object of a reduced channel. For a family of completely positive trace-preserving maps (\Phi_t), we set (B(t)=J(\Phi_t)/d_S) and study the thresholded Choi-support layer determined by (P_\tau(B)), its effective rank (r_\tau(B)), and the threshold margin (g_\tau(B)). A deterministic tolerance (\eta) defines the certification rule: if (g_\tau(B)>\eta), the threshold-rank layer is stable under operator-norm perturbations of radius (\eta); if (g_\tau(B)\le\eta), the continuation status is MARGIN_LOW, and a pathwise continuation of the previous support certificate must record CERTIFICATE_RESET before proceeding. The protocol is demonstrated on an exactly solvable two-spin central-spin dephasing model. In the selected parameter regime, the coarse Markovian envelope distance is nearly zero at the very time when the Choi threshold-support certificate fails. Thus small coarse model discrepancy does not imply stable thresholded support certification. The construction does not reconstruct the physical bath, identify a true hidden environment, equate Choi rank with physical bath dimension, or interpret certificate reset as a physical event. It provides a deterministic finite-dimensional diagnostic for reduced-channel data under declared thresholds and tolerances.https://github.com/AIDevelopersMonster/boundary-compensationRelation: isSupplementedBy / isSupplementTo, depending on Zenodo field wording
This preprint develops the theory of spectral closure systems: an operator-algebraic framework for local access to a global compatibility object through typed, lossy closures, organized as a calculus of admissible sections of access maps. Truth is extendability: a local datum is globally valid exactly when it extends along the inverse system, with finite-stage obstruction certificates by compactness. Recovery of local charts is canonical exactly on the sufficiency sector (Petz recovery, conditioning); off it, every totalization is impotent or pays an exact relative-entropy price. Time is the Radon–Nikodym cocycle of the failure of a nonsingular dynamics to preserve belief, realized as the modular flow of its crossed-product envelope; a nontrivial clock forces a noncommutative envelope, with Murray–von Neumann–Connes type read off the ratio set and the flow of weights as envelope invariant. In the quantum perimeter, point-valued global truth is empty (Kochen–Specker), state-valued compatibility survives (Gleason, Christensen–Yeadon), Bell violations measure the classical gluing defect (Fine), and stable classical recordability is characterized as the commutative, modular-invariant, asymptotically sufficient chart sector; on the reflection-positive sector the record process reconstructs a positive-generator dynamics. Globality is relative but closure is terminal: stage sets collapse inward under directed exhaustion, the envelope tower closes outward at the second step by Takesaki duality, unforced finite-entropy residuals decay at a rate uniform in the modular cocycle (a complete logarithmic Sobolev inequality by transference from the circle), and the forcing cost balances exactly in dilation against system–environment correlation. Further results include an independence theorem for the underlying skeleton, a two-tier classification theorem with provable Foreman–Rudolph–Weiss wildness inside its fibers, an envelope universality theorem (the temporal envelopes are exactly the injective von Neumann algebras carrying a Cartan subalgebra), and a sector-split Orlicz envelope for relative entropy beyond bounded densities. Every external input is a named theorem-import with matched hypotheses; the interpretive vocabulary is proved to be a conservative definitional extension of the formal core.