Certifying unused optical states with imperfect two-copy interference

What can an imperfect destructive optical test guarantee about the light that remains unused? This article gives an explicit certificate of proximity to coherent states for the unmeasured members of an independent, possibly nonidentical batch. Its main optical contribution is a physical symmetrization whose error can be controlled by mean photon number alone, while retaining every photon-number sector. Cancel the optical error before estimating it Let \(P_e\) be the input-picture projector for vacuum in every relative output mode of a beam splitter at angle \(π/4+e\). The same port rotation acts on each paired internal mode. A fair random exchange of the complete effective inputs, after arm-specific passive transformations, replaces the event by \((P_e+P₋ₑ)/2\). For any normal two-arm state \(Ω\) with total mean photons at most \(E\), and \(|e|≤θ≤π/4\), \[|TrΩ({P_e+P₋ₑ}{2}-P_0)|≤1+√3/2Eθ^2.\] The exact projector curvature on the total \(k\)-photon sector is \(k+√3k^2-2k\). Its linear growth in \(k\) permits the infinite-tail return using a first energy moment, without a photon cutoff or a mode-count factor. The article also charges imperfect randomization and distinguishes passive mismatch from angular bias. A guarantee for the actual untested preparations Prepare \(B\) pairs in the product state \(ᵢ₌₁B(ρ_i⊗σ_i)\): all \(2B\) preparations are independent, although their states may differ and may be mixed. Each pair obeys the total mean-photon bound \(E\). Select a uniform fixed-size sample of \(n<B\) pairs, independently of the batch and its potential test rates, including after conditioning on valid calibration information. Let \(U\) be the remaining indices and \[D(ρ)=inf_α\|ρ-|αα|\|_1, D_U\,2=1/2(B-n)∑i∈ U[D(ρ_i)^2+D(σ_i)^2].\] The norm is the full trace norm, with maximum distance two. For a predetermined zero-click protocol, observing no recorded clicks gives the frequentist certificate \[ D_U\,2≤min\{4,4c[δ+νη+{Bln(1/βsig)}{η n(B-n)}]\}.\] Here \(δ\) bounds optical inflation of the passing probability, \(η>0\) is the retained detection-efficiency lower bound, and \(ν\) is a separate absolute click-probability decrement. The coefficient is \(c=1-exp(-4/ C)\), where \( C≥ C_\) comes from the established sharp Wehrl theorem; the displayed examples use \( C=4.114205\). Total failure probability is at most \(βcal+βsig\). The paper also supplies bounds for nonzero counts and a separately scoped, anytime-valid result for tested conditional streams. Resources, calibration and scope The source requirement is a product batch, not merely product structure conditional on an unobserved shared label. The conclusion controls average squared distance to the coherent-state set at the retained state plane; different preparations may have different coherent references. Detector bounds must cover all relevant modes and multiphoton inputs. Single-photon calibration estimates efficiencies within that trusted response model. Signal consumption is \(2B\) preparations, with \(2n\) tested and \(2(B-n)\) retained. Calibration is additional. A fully specified hypothetical single-mode example uses three million incident calibration photons and 20,000 signal preparations, with at most 3,200,000 mean incident photons in total. Source-generation attempts, control and storage costs are additional. The figures evaluate proved bounds and certificate floors; they do not report experimental measurements, demonstrated hardware performance or universal impossibility limits. The analysis builds on established coherent-state comparison, quantum sampling and the sharp Wehrl entropy-to-distance theorem. Its contribution is the full-Fock optical error control and its compatible return through calibration and counting to the actual unused product batch. The integrated 14-page article contains two figure groups, complete proofs and references.