Finite-Count Calibration of Ancilla-Orientation Safe Maps A Bell-Apparatus Bound and a Reference-Link Obstruction
Year: 2026
Type: preprint
Abstract: An exact operational selector specifies which candidate maps are nonnegative on declared preparation and effect resources. This paper studies finite-count calibration of that resource-dependent question for a qubit apparatus with an inert qubit reference, an arbitrary complex physical preparation, and an arbitrary physical binary detector, relative to a trusted common calibration frame. Two preparation-stabilizer settings and six separable detector probes control the statistic of the affine family (1 − θ)id + θT, 0 ≤ θ ≤ 1, uniformly over all compatible state–effect pairs. The explicit radius is 3h + √(2s), where h is a simultaneous detector calibration width and s is a one-sided bound on Bell infidelity. It yields simultaneous inner and outer confidence sets for a one-parameter slice of the declared safe-map class. A fixed allocation of 200,000 trials per calibration setting, totaling 1,600,000 calibration trials, gives false-implication probability at most 0.005 and transpose-exclusion power at least 0.996 at the specified physical alternative. The calibration confidence budgets are 0.001 for preparation and 0.004 for the detector. This exclusion calculation uses no target stream; the acquired calibration circuits can use identity wiring. Coverage and prospective power remain separate statements. The acquisition contract uses fixed-length iid Bernoulli streams with their stated physical means and permits coupling between whole streams. The numerical example sets external-reference systematic errors to zero, and its trial budget excludes establishing reference trust and accuracy. If the preparation and detector references are not linked, the paper constructs two physical identity-wired apparatus worlds with identical complete record laws under the specified independent acquisition construction but different safety conclusions on the fixed target cone pair. A secondary nearest-center theorem supplies unconditional error control for the joint event of a wrong and nonabstaining candidate label under an additional finite-context target-law contract. Its nonidentity candidates are not completely positive maps, and their ideal probabilities lie below the nominal detector’s ordinary positive-output floor. That target law is an explicit model assumption; the label theorem is not a conditional-correctness or posterior statement. The results are analytic and prospective. The number of settings and trial allocations are sufficient constructions, with no empirical success-rate, minimality, optimal sample-complexity, or general certified count-to-feasibility implementation claim. The scope is one declared qubit apparatus, target preparation ray and binary effect cone, and affine map slice. The common complex operator description, trusted reference relationship and physical calibration probability law are assumptions. Exact finite checks and an assessed supporting Lean subset accompany the manuscript. That subset covers 326 audited named declarations: 240 theorems and 86 definition/type interfaces. Analytic binomial coverage, concentration and acquisition bridges, independent-product-law construction and normalization, and the complete noisy-power chain remain outside the accepted Lean subset. That subset does not establish a certified numerical feasibility evaluator, a complete physical two-sector implementation, or an ordinary-channel realization. AI assistance and the dated review/correction provenance are disclosed in the manuscript and companion records. No experimental dataset, external human peer review, or end-to-end formal verification is claimed.
Author: Zhang, Qingchun
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225571308