Flagged Unitary–Replacer Channels: Sector-Resolved Choi-Moment Spectral Completion, Fixed-Fidelity Choi-Entropy Fibers, and Joint Transparency–Spectrum Identifiability

Abstract

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.