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.
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.
This is a separate preprint record that develops the robustness layer of the outcome-resolved mid-circuit-measurement gauge analysis. It uses the exact complete-positivity benchmark of the preceding preprint as an input, but its main result concerns the composition of imperfect-control uncertainty, residual-gauge conditioning, and instrument-model error. The general ingredients—gate-set-tomography gauge freedom, commutant geometry, perturbation by singular-value gaps, quantum-instrument physicality, and mid-circuit-measurement learnability—are prior art. The paper does not claim a new general GST gauge principle, a dimension-free tomography theorem, or experimental validation. Related identifier ● Identifier: 10.5281/zenodo.23043202 ● Relation: Is derived from (isDerivedFrom) ● Resource type: Publication -> Preprint ● Title: Outcome-Resolved Mid-Circuit Measurements as Exact Gauge Constraints: Complete-Positivity Windows, Correlation Certificates, and Robustness Limits Description / Abstract Self-consistent quantum characterization is invariant under similarity transformations, so an experimentally fitted gate set is generally identified only up to gauge. Exact trusted controls can reduce this ambiguity, but realistic controls are imperfect; simultaneously, a mid-circuit measurement (MCM) instrument is itself only approximately described by a model. These two uncertainties act at different logical stages and should not be merged before the gauge geometry is reconstructed. This preprint gives an explicit finite-dimensional stability analysis. For a finite reference control family ({R_i}), define the commutator map (\mathcal C(B)=([B,R_i])_i) and let (\gamma) be its smallest positive singular value. If two exactly gauge-equivalent implemented gate families ({G_i}) and ({G’_i}) both lie within Frobenius errors (\delta_i) of the same references, then every trace-preserving gauge matrix (B) lies within [\operatorname{dist}{F}(B,\ker\mathcal C)\le \frac{2\Delta}{\gamma}|B|{\mathrm{op}},\qquad\Delta=\left(\sum_i\delta_i^2\right)^{1/2}.] For the two-qubit control set (H\otimes I), (S\otimes I), (I\otimes H), (I\otimes S), and CNOT, the exact commutant is (\operatorname{span}{\mathcal D,\mathrm{id}-\mathcal D}), so the residual scalar gauge is (B_c=\mathcal D+c(\mathrm{id}-\mathcal D)). In the declared unweighted Frobenius convention, direct computation gives (\gamma=0.830129650657\ldots). The main result closes two previously separate robustness budgets. Suppose an independent physical or calibration argument places the scalar projection (c) in a positive prior interval ([\ell,u]). If (\alpha=2\Delta/\gamma<1), a self-normalized estimate removes the otherwise unknown factor (|B|): [|B-B_c|_F\le \bar e=\frac{\alpha M_0}{1-\alpha},\qquad M_0=\max(1,u).] If (\bar e<m_0=\min(1,\ell)), the condition number of (B) is explicitly bounded. Composing this control-side bound with a Choi-Frobenius instrument-model error (\varepsilon) yields a closed outer complete-positivity condition for the symmetric noisy Lüders MCM benchmark, [(1-\eta)|1-2p|,|c-c^{-1}|\le \eta+2d\rho,] where (\rho) is an explicit function of (\Delta,\gamma,\ell,u), and (\varepsilon). Hence the imperfect-control problem again reduces to an explicit scalar gauge interval. The result separates transverse control uncertainty from instrument-model uncertainty and identifies the loss of robustness as the prior gauge envelope approaches singularity.
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.
Accurate modeling of midcircuit measurements (MCMs) is essential for dynamic-circuit operations such as syndrome extraction, measurement-based reset, and the separation of state-preparation and measurement error. Unlike terminal measurement, a noisy MCM both produces a classical outcome and alters the incoming quantum state, thereby influencing subsequent circuit operations. This makes conventional confusion-matrix or fidelity-level characterization insufficient. Here, we introduce an efficient, self-consistent protocol for learning a single-qubit Z-twirled MCM instrument, retaining the readout-backaction correlations and excitation-decay asymmetry that are erased in Pauli-error descriptions. Remarkably, readout bit strings from only three repeated MCMs on a maximally mixed input determine all learnable parameters of the reduced instrument, up to a single unidentifiable gauge degree of freedom. Physicality constraints convert this nonidentifiability into narrow, gauge-aware error intervals. Implemented on IBM superconducting processors, the learned instrument improves Pauli-observable prediction by ∼100× over a conventional confusion-matrix model and reveals a T1-decay dominated backaction. Our protocol provides a compact characterization layer for state-preparation and measurement error separation, reset optimization, and noise-aware quantum error correction.
Physical Sources, Finite Detectors, Persistent Records, and Actual Events This paper develops a unified account of quantum measurement under the Admissibility and Standing Constraint (AASC) framework, connecting constructed physical sources, quantum instruments, material amplification, finite record retention, event probabilities, and the actuality of measurement outcomes. The paper distinguishes three separate objects: a complete quantum instrument containing all outcome operations, including unresolved and failed outcomes; a material record carried by a physical system whose acquisition and retention can be analyzed; an actual history describing which occurrences belong to one original experimental run. A complete instrument and a persistent material record can be constructed without assuming that linear quantum dynamics alone selects one globally definite original-run history. Conversely, the admission of an actual event does not by itself determine its probability law or conditional quantum state. The physical construction contains two complementary measurement targets: a regulated interacting gauge–Higgs–fermion source with its full source Hamiltonian retained during finite receiver pulses; an explicitly parameterized six-mode cosine circuit with source and pump interfaces, losses, distinguishable outputs, a two-polarization intrinsic-spin receiver, and a finite energy-resolving thermal amplifier. The material-registration chain includes source-code loading, driven conversion, output registration, pump-displacement removal, transfer to an intrinsic material spin, finite Gibbs-collision amplification, endpoint reading, and quiet record retention. Electronic display labels, material flags, missed counts, dark displays, blanks, and unreadable outcomes remain distinct and are retained in the joint error ledger. The paper establishes: informative bounded receivers on an interacting quantum source; finite-pulse calibration with the complete source Hamiltonian retained; a complete monitored detector model; finite preparation and amplification bounds; coherent composition of instruments and records; adaptive record composition; local-instrument compatibility; calibrated affine probability uniqueness; sharp coherent calibration limits; sequential probability laws and compatible conditional states; explicit actual-history requirements. A finite Fourier residual certificate bounds the complete monitored detector instrument over its full 35.587ms interval by 0.063241 in restricted diamond norm. Exact interval arithmetic gives seeded amplifier endpoint fidelity greater than 0.9999235242 after 1,024 finite collisions, together with an explicit conditional survival bound for a subsequent 100ms quiet hold. The probability analysis shows that positivity, density-operator dependence, affinity under classical mixtures, and calibration fix the trace probability rule for the declared readout. Approximate sector calibration alone permits coherent deviations of order the square root of the calibration error; stronger linear bounds require additional structure. The paper’s central measurement-theoretic conclusion is deliberately separated from its successful detector construction. A complete instrument admits an exclusive-history completion, but linear quantum dynamics alone does not select one such history. Source-specific preparation, reader analysis, and a bounded counting-law completion supply the additional physical input required to select an actual history. The resulting framework substantially develops the material and probabilistic parts of quantum measurement while isolating apparatus errors, actual-history law, and event-selection assumptions as explicit obligations. The work does not claim to solve every interpretation of the measurement problem, derive a universal collapse law, establish unrestricted detector accuracy, or identify a finite calibration certificate with an experimental calibration. Principal contributions A unified source-to-detector account of quantum measurement. Constructed interacting quantum sources with finite receivers. Six-mode cosine-circuit detector realization. Material registration and finite amplification chain. Two-polarization intrinsic-spin receiver. Finite thermal collision and endpoint-reading model. Full-duration restricted diamond-norm detector bound. Interval-certified amplifier fidelity. Conditional quiet-hold survival bound. Retention of blanks, wrong, unreadable, and missed outcomes. Calibrated affine probability uniqueness. Sequential instruments and compatible conditional states. Explicit separation of complete instruments, material records, and actual histories. Source-specific conditions for actual-event selection. Complete physical error and assumption ledger.
Exact identification of a target instrument need not imply a linear stability bound for nearby observations. We study fixed finite-dimensional complete instruments with rank-one target Choi blocks, all outcomes retained, a fixed diagnostic contamination path with strictly positive branch noise, and an all-device bound on Choi mass outside the target rays. Under these hypotheses, linear prescribed-target certification is equivalent to injectivity of the joint observation and completeness map on the physical target-tangent space. An invisible nonzero tangent yields an exactly feasible positive instrument pair with square-root separation; a separate target-ray condition distinguishes sharp square-root behavior from constant-order ambiguity. For two prescribed three-basis readouts on a four-dimensional incidence-record benchmark, we obtain linear and square-root compatible-instrument radii, respectively. We then distinguish arbitrary probability residuals from simultaneous physical preparation and readout errors. Positivity and normalization give a support-amplitude estimate under which both protocols have linear calibration-only sensitivity, although their responses to contamination and unrestricted probability error differ. A fixed-sample application uses additional exactly trusted references to acquire operator-error bounds and obtain a simultaneous full-instrument confidence statement. A separate named-source power calculation reaches the prescribed error threshold with a conservative sufficient allocation of approximately 6.12 × 10^21 trials. The results concern a fixed laboratory-frame target and supplied common-device and sampling assumptions; they establish neither practical efficiency nor device-independent calibration. Written proofs are supported by exact finite certificates and a returned kernel-checked Lean subset; the full argument is not formalized end to end.
Gate Set Tomography determines experimental probabilities only up to similarity transformations of the fitted gate set. In practice, reported error metrics are therefore evaluated after choosing a gauge representative, often with gauge fixing restricted to unitary transformations.This work studies the geometry of the larger physical gauge orbit and asks whether nonunitary but still physically admissible representatives can improve reported gate-quality metrics beyond unitary gauge fixing.Analytically, the paper identifies a class of unital common-isotropic models for which nonunitary gauge transformations provide no advantage, and shows how a quadratic advantage appears when common nonunitality is introduced. Exact results are also derived for pure-translation gauges, together with a gauge-invariant lower bound.The analytical mechanism is tested numerically in two complementary settings. First, a historical GST model is audited conditionally, with the model-mismatch limitation stated explicitly. Second, a synthetic CPTP benchmark is generated, refitted by maximum likelihood, and shown not to be rejected by its goodness-of-fit test. In that well-specified setting the nonunitary gauge advantage reappears and agrees with the analytical prediction to numerical precision.A paired parametric bootstrap shows that the absolute value of the reported error metric can fluctuate substantially under sampling while the difference between unitary and nonunitary gauge fixing is much more stable because the two estimates are strongly correlated. An analytical bound also quantifies the maximum relative size of the nonunitary advantage within the solvable model family.The archive contains the manuscript, numerical certificates, reproducibility scripts, provenance information, package manifests, and automated verification tools. Version 1.8.1 differs from version 1.8 only by a clarification in the smoke-test output; the manuscript and scientific results are unchanged.