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.
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.
Self-consistent quantum tomography is invariant under similarity transformations that can leave all observed circuit probabilities unchanged. Mid-circuit measurements (MCMs) add a qualitatively different object to this problem: an outcome-resolved quantum instrument has both a classical label and a post-measurement quantum state, so each outcome branch must itself remain completely positive under any admissible representation change. We study an exactly solvable residual-gauge model in even dimension d = 2r ≥ 4. After a fixed two-qubit Clifford control set reduces the trace-preserving similarity freedom to the scalar family B_c = D + c(id − D), we consider a symmetric two-sector Lüders instrument with label-confusion probability p and isotropic state-and-label replacement probability η. For the transformed branches J_{s,c} = B_c J_s^{η,p} B_c^{-1}, we prove exact necessary-and-sufficient complete-positivity criteria for both positive and negative scalar gauges. For c > 0 the condition is (1 − η)|1 − 2p||c − c^{-1}| ≤ η, while for c < 0 it is (1 − η)|1 − 2p|(|c| + 2 + |c|^{-1}) ≤ η. The result follows from an explicit Choi-spectrum calculation. It yields a closed positive gauge interval, an exact threshold for the appearance of negative gauges, and a logarithmic gauge radius |log c| ≤ arsinh(K/2), with K = η/[(1 − η)|1 − 2p|]. If the outcome label is discarded, the averaged channel is invariant under every B_c, so the additional gauge information disappears exactly. Two lag correlations determine K within the assumed model and therefore give an observable conditional gauge interval; a Hoeffding bound gives a finite-sample outer interval. We also derive an explicit outer interval when each instrument branch is known only within a Choi-Frobenius error budget. Finally, a Lüders-versus-measure-and-prepare countermodel shows that repeated outcome strings alone do not validate the quantum state continuation. A targeted literature audit finds substantial prior art on gate-set-tomography gauge freedom, instrument tomography, MCM learnability, and gauge-aware physicality intervals, including a closely related 2026 three-measurement protocol. The contribution is therefore deliberately narrow: the exact higher-dimensional full-complete-positivity boundary for the stated instrument family and its derived consequences, not the general idea that physicality can restrict tomography gauge.
We establish a global, worst-case non-identifiability theorem for periodically driven finite-dimensional quantum systems. On the unrestricted smooth periodic Hamiltonian class, we show that the period-averaged Fubini-Study metric component of a fixed initial state cannot, in general, be reconstructed from exact one-period propagators indexed by every starting time and external parameter. Thus, even complete starting-time-resolved endpoint data are insufficient to determine this intra-period geometric quantity. The obstruction is characterized exactly. The starting-time-indexed endpoint data determine the conjugation path of the monodromy, but not its particular unitary lift. On each fixed-monodromy slice, the observational fibres are precisely the right orbits generated by smooth parameter-dependent based loops taking values in the pointwise centralizer of the monodromy. The Fubini-Study functional is not invariant under this fibre action and therefore does not factor through the endpoint observation map. An explicit real-analytic two-level witness demonstrates the obstruction within a commuting, one-generator Hamiltonian family, so neither non-Abelian time ordering nor Floquet-logarithm ambiguity is required. A continuous family of Hamiltonians produces identical starting-time-indexed one-period endpoint data while yielding different, and on the unrestricted class arbitrarily separated, period-averaged Fubini-Study geometry. Non-identifiability persists under any prescribed uniform bound on the parameter derivative of the Hamiltonian. The result is a deterministic global statement, not a claim of generic non-identifiability or experimental impossibility. It identifies a precise information gap between complete one-period endpoint data and full intra-period dynamics: the missing information is the unitary lift of the observed monodromy path rather than ordinary scalar phase freedom.
The spectral measure of a Hamiltonian and the POVM implemented by an apparatus need not coincide. We formulate observable attribution relative to an apparatus class defined by its preparations, controls, ancillary couplings, memory, and readable records. For a nondegenerate qutrit coupled to a binary threshold pointer and an amplifying environment, the explicit dynamics induces a classical observer channel that preserves only ground-versus-excited weight. We prove complete factorization of every allowed primitive through this channel and lift it to all finite adaptive protocols. Thus repetition, coherent controls within the excited sector, sector-only ancillas, calibration, and feed-forward cannot distinguish the two excited eigenstates. The full energy PVM is not attributable to the base class, and its optimal worst-case total-variation error is exactly one half. Adding a selective sector-changing control realizes the full PVM. Within the specified finite-protocol resource model, its exact minimum integrated operator-norm action is ; the optimal one-pulse approximate cost is also obtained in closed form. For imperfect devices we derive an exact minimax decoder for arbitrary calibrated binary responses, explicit error floors from record overlap, leakage, detuning, and dephasing, and a diamond-norm stability theorem for finite adaptive protocols. Observable attribution is therefore determined by physically available channels and resources, although the outcome statistics of every fixed experiment remain objective. This is a preprint version of a manuscript currently under journal review.
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.
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.