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Finite-Count Power for Target-Centered Instrument Certification

Qingchun Zhang

Published Oct 31, 2026• 0 citations
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Abstract

RTCG-P5, version 1.0, establishes finite-count passing guarantees for a target-centered certificate of a prescribed quantum instrument on a specified family of imperfect preparations. Six rank-two states share one signed, support-preserving parameter epsilon in [-10^(-6),10^(-6)]. The diagnostic instrument is fixed; actual readouts and trusted references remain nominal. Both the new rule and the old scalar rule use the same acquired preparation allowances. An explicit doubled fixed allocation of 636776064000000000000 attempted trials yields the upward-reported sufficient upper 0.000098454319447, strictly below 10^(-4). For every fixed parameter in the family and every allowed coupling between settings, the new rule passes with probability at least 1 - 873 exp(-12), approximately 0.994636102615544. Trials are iid within each setting; independence across settings is not assumed. The original allocation's sufficient envelope is 0.000108332505187 and does not close. This does not establish failure of the actual rule or a necessary acquisition cost. A separate single-analyzer argument proves an actual-output obstruction for the unchanged scalar preparation-penalty rule. At either signed endpoint, its reported upper is at least 0.00672 except on an event of probability at most 2 exp(-12), for every fixed valid allocation retaining that conversion. The obstruction is specific to this rule. It is not an information-theoretic lower bound or a lower bound on the physical instrument error. The universal false-certification bound, 889 exp(-12), remains a separate inherited guarantee for complete complex CP competitors under the stated trusted-reference and shared-world contract. The family-specific power guarantee does not restrict that competitor class, and the two probability statements are not combined into one 99-percent event or a posterior correctness assertion. The manuscript and mathematical supplement provide the physical construction, concentration proofs, all signed response coefficients, whole-interval argument and exact arithmetic. The companion provides editable sources, byte-preserved numerical programs and data, supporting Lean sources, actual execution/type/axiom evidence and reader instructions. The retained Lean 4.19.0 execution supports the stated deterministic and conditional probability implications. Physical-source realization, concentration/Holder arguments, inherited unrestricted physical coverage, complete raw-count linkage and general Python refinement retain their explicit formal boundaries. Packaging the public release is not a new Lean execution. This is a new finite-count extension of RTCG-P4, Target-Centered Preparation Calibration for Outcome-Resolved Instrument Certification (10.5281/zenodo.23789860). No experimental acquisition, practical sample efficiency, fewer-shot result or optimal acquisition cost is claimed. Manuscript, supplement, documentation and author-owned mathematical data use CC BY 4.0; original numerical, checking, build and Lean code use MIT. Inherited and third-party notices remain authoritative for their material.

DOI
10.5281/zenodo.23838975

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RTCG-P4, version 1.0, is a Complementarity-First preprint on using signed preparation-calibration responses to sharpen a uniform certificate for a prescribed complete quantum instrument. Fifteen trusted Pauli analyzers on each of six preparations determine the target contribution to a displaced-input validation response. Complete-flag trace cancellation bounds the remaining unknown-instrument contribution by an acquired preparation allowance times the actual unhalved flagged diamond error. An inherited source-specific instrument estimate then yields a record-based maximum-of-ratios bound with explicit denominator, cap, and fallback conditions. The same-record synthetic count comparison improves from 0.000079105273917 to 0.000071209076256. A support-preserving preparation-error population example gives the new sufficient upper 0.000074611876587 versus the inherited 0.008404839640705. The latter is a population comparison; it does not establish nonnominal finite-sample power. Both certificate rules hold on the same confidence event. Named-source power retains nominal actual calibration, and the fixed acquisition schedule remains unchanged. The written proofs allow arbitrary complete complex CP competitors under the stated trusted-reference, shared-world and within-setting iid sampling contract, with dependence between settings allowed. Exact finite source and implementation evidence is provided. A source-matched Lean companion supports the deterministic response and conditional instrument-composition chain. Physical acquisition/source estimates, the complete-channel cap, identification of the full compatible family, stabilization/representation, and probability/count implementation retain their explicit formal boundaries. The public materials are the main manuscript, supplement, and selected reproducibility companion with editable LaTeX, exact numerical inputs and code, accepted Lean sources, historical execution evidence and verification instructions. Manuscript, documentation and author-owned data are licensed under CC BY 4.0; original numerical, checking and Lean code are licensed under MIT. Retained inherited and third-party notices continue to apply.
We study how selected physical calibration data can retain a uniform guarantee for acomplete quantum instrument. For a fixed four-dimensional Hadamard projective measurement,six trusted real pair-superposition probes supply forbidden-event leakage D and one unaveragedoutcome-weight sum W , with nominal value three. Zero forbidden probabilities alone leave a phys-ical effect-weight ambiguity. Adding the weight constraint identifies the target, and positivity withshared POVM completeness gives the robust maximum-effect bound min{1, (W−3)+ +3D/2+√D}for arbitrary-rank complex effects. The linear weight allowance improves the local use of ordinaryfinite-count weight statistics relative to a coarse square-root conversion. Combining this result withtwo inherited readout calibrations gives a 4 + 6 + 6 added-readout architecture within the exist-ing trusted menu. We expose its composition with the inherited support, positive-Choi-remainderand complex-tangent interfaces for an unhalved flagged-diamond guarantee. A fixed-count rulehas uniform false-certification risk below 0.01 and a separate passing probability above 0.99 at thedesignated source with nominal actual preparation and readout, assuming iid trials within settingsand allowing dependence across settings. On the named-source power event, an exact sufficientallocation of 3.18388032 × 1020 attempts gives a reported upper at most 0.000094848998413 < 10−4.This is a source-specific construction and sufficient resource comparison; it does not establish op-timal sample complexity or experimental practicality. Written arguments are supported by exactfinite source and implementation checks. A companion Lean development verifies the deterministicsix-probe bound, population-ceiling transfer and exact-identification conclusion.
The observations available for a quantum instrument need not guarantee the accuracy of the device that generated them. On a fixed four-dimensional incidence-record benchmark, the paper compares two readout augmentations retaining six mixed preparations, both instrument outcomes, all attempts and shared completeness. Each adds three four-outcome bases and eighteen preparation/readout pairs. At r = 1/300000000, the 123 design admits a complete compatible instrument with unhalved prescribed-target diamond error at least 0.000103738856511, above the prospective 10^(-4) threshold. Replacing one basis gives the 124 design and a uniform upper 396319/8640000000 = 0.0000458702546296296... for every compatible complete instrument. The diagnostic generates data; its noise law and a Choi-rank bound are not imposed on the unknown device. The added basis effects observe ten Hermitian output directions, while original destination tests supply the leakage control. Five triples in the fixed finite menu are sufficient, one has a complete counterexample, and four remain unresolved. The paper claims neither a general measurement minimum, optimal radius, experimental certification, finite-sample power, calibrated hardware nor a general small-noise rate classification. Exact instrument identification remains obstructed on the unchanged preparations. Scientific cutoff: AUG04, 15 October 2026; AUG05 remains unexecuted. The companion contains editable manuscript sources, complete exact scientific arrays, the unchanged standalone reader, and accepted returned Mac Lean evidence for a supporting subset: 83 supporting theorems, 37 definitions and one abbreviation, with all 121 source/type/axiom correspondences. The complete-instrument theorems are not formalized end to end. Public execution views have documented path-only redactions; this preparation does not add a kernel run or external human peer review.
Calibrated support observations can constrain a complete quantum instrument without identifying its action by unrestricted tomography. We study what those observations guarantee when the noise law is not supplied. On a fixed four-dimensional incidence-record experiment, we define the set of all complete two-outcome completely positive instruments compatible with the exact raw rejection and outcome probabilities, retaining the classical flags. Exact ideal data identify the target, but an explicit family of complete instruments reproduces the nonzero-data category laws of a low-error contamination device while admitting coherent error of square-root order. A target-ray mass certificate and a common complete-dilation comparison give a universal upper, while an exactly feasible rational instrument supplies a stronger lower witness. At the diagnostic parameter r = 1/300000000, the unhalved target-relative worst-case radius lies between 0.000258065115532250 and 0.000951566242566. The result separates avoidable inequality conservatism from information absent from the measured data. Fixed-functional and fixed-map optimizations have separately certified scopes; they do not solve the global radius. The conclusions are conditional on the supplied carrier, calibration and common-device model. They do not establish an apparatus, device-independent certification, a finite-sample power guarantee or complete formal verification. Selected supporting declarations have one recorded successful Lean 4.19.0 macOS execution and declaration-specific source, type and axiom review; this is not an end-to-end formalization.
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.
We construct a source-matched reference calibration for distinguishing the coherent law h = 0 from the specified added generator h = J/25 in a finite quantum-record model. A complete-carrier transfer bound connects an accessible reference contrast to a delayed target contrast while retaining the original drift and actual law. On reference coverage and measured admission, restricting both hypotheses by the same observed reference band yields a directed target-set separation of at least 0.0071230619 whenever both compatible classes are nonempty. Rival worlds may have different admitted nuisance parameters. The constants are uniform over independently known finite lengths m >= 65, with tag 65, access prefix 1:32 and target deadline 25.02/J. Finite reference preparation and work/source reset satisfy the declared response budgets. Reset acts on the actual joint input and consumed supply, with the retained reference R exactly inert. It does not assign a universal reduced channel to correlated supplies. Under almost-sure conditional resource and supply bounds, the comparison extends to history-dependent acquisition. A prespecified rule, including abstention below the calibration threshold, has each-law unconditional wrong-declaration probability at most 2 exp(-6) + exp(-85/16) < 0.01. This is neither a posterior probability nor an error bound conditional on admission. The statistical step uses standard bounded-martingale concentration. Global target states and specified auxiliary supplies remain resources. Reset requires additional globally conditioned controls and has very large finite costs. The result is a conditional theorem for the two declared laws; it does not establish practical implementation or a unique diagnosis of fundamental collapse. A supporting Lean 4 companion contains 85 theorem declarations and 30 supporting definitions across Calibration, Operations and Risk. It provides checked scalar, sufficient-bound, resource-arithmetic and finite-history risk consequences under explicit premises. The companion has an executed Linux preflight and accepted returned macOS compilation, named-type and transitive-axiom evidence. The quantum constructions and conditional moment, filtration, tower and tail derivations remain written proofs or external premises; this is not an end-to-end formalization of the physical theorem chain. The original checked companion targeted author v0.4 and is carried forward unchanged, with its scope and execution chronology preserved.