Can a measurement that records only one global vacuum event recover an optical mode spectrum? This article identifies when increasing the number of simultaneously interfering copies supplies exactly the spectral information lost by a two-copy test. It provides a finite algebraic reconstruction without first determining the generating mode basis. Source and measurement. Consider a pure source with known total photon number \(N\ge2\) and at most \(m\) occupied orthogonal modes: \[|\Psi\rangle=\sum_{j=1}^{m}\sqrt{w_j}e^{i\theta_j}|N_j\rangle,\qquad w_j\ge0,\qquad\sum_jw_j=1.\] Here \(|N_j\rangle\) has all photons in mode \(j\), and the target is the unordered spectrum \(\{w_j\}\) of the normalized one-particle reduced state. The source phases and a common passive mode basis may be unknown. This source class includes every pure two-photon state; for \(N\ge3\), the generalized NOON/GHZ form is an explicit preparation promise. Interfere \(\ell\) complete, independent, identically prepared copies in a Fourier multiport, acting identically on every physical signal mode. The probability that all relative output modes are vacuum is \[M_\ell=\ell^d\int Q_\Psi(z)^\ell\,d\mu_d(z),\qquad Q_\Psi(z)=|\langle z|\Psi\rangle|^2,\qquad d\mu_d=\pi^{-d}d^{2d}z,\] where \(d\) is the physical mode number. Spectator vacuum modes do not change this probability. Exact reconstruction and its mechanism. The probabilities \(M_2,\ldots,M_m\), together with normalization, determine every mode weight, including zero and repeated weights. The new connection is from these optical probabilities to spectral power sums. A formal logarithm separates connected photon pairings; their strictly positive coefficients make the map triangular and invertible at every order. Standard Newton identities then recover the spectrum. For generic positive, distinct weights, the lower prefix through \(M_{m-1}\) leaves even the largest weight undetermined. A concrete pair of three-photon sources has spectra \((1/2,1/2,0)\) and \((2/3,1/6,1/6)\). Both give \(M_2=11/64\), while their three-copy probabilities are \(155/6561\) and \(183/6561\). The reconstructed largest weight also determines the coherent peak and full trace-norm distance to a pure coherent state: \[q=\frac{N^Ne^{-N}}{N!}\max_jw_j,\qquad D_{\mathcal C}^{\,2}=4(1-q).\] Physical resource accounting. Highest interference order is distinct from total sample consumption. With \(n_\ell\) independent trials at order \(\ell\), the total number of input copies is \(C_{\rm in}=\sum_{\ell=2}^{m}\ell n_\ell\), and the incident photon count is \(NC_{\rm in}\). A finite two-hypothesis counting example is included. The exact inverse is not a uniform noisy-spectrum confidence theorem, and the order-necessity result concerns this specified vacuum-event hierarchy rather than arbitrary optical measurements. Mode-resolved one-particle measurements provide a different route to the spectrum. The integrated 13-page article contains two figure groups, complete proofs and references. It distinguishes the all-order optical-to-spectral connection from established replica-moment formulas and classical power-sum inversion.
Can a measurement that records only one global vacuum event recover an optical mode spectrum? This article identifies when increasing the number of simultaneously interfering copies supplies exactly the spectral information lost by a two-copy test. It provides a finite algebraic reconstruction without first determining the generating mode basis. Source and measurement. Consider a pure source with known total photon number (N\ge2) and at most (m) occupied orthogonal modes: [|\Psi\rangle=\sum_{j=1}^{m}\sqrt{w_j}e^{i\theta_j}|N_j\rangle,\qquad w_j\ge0,\qquad\sum_jw_j=1.] Here (|N_j\rangle) has all photons in mode (j), and the target is the unordered spectrum ({w_j}) of the normalized one-particle reduced state. The source phases and a common passive mode basis may be unknown. This source class includes every pure two-photon state; for (N\ge3), the generalized NOON/GHZ form is an explicit preparation promise. Interfere (\ell) complete, independent, identically prepared copies in a Fourier multiport, acting identically on every physical signal mode. The probability that all relative output modes are vacuum is [M_\ell=\ell^d\int Q_\Psi(z)^\ell,d\mu_d(z),\qquad Q_\Psi(z)=|\langle z|\Psi\rangle|^2,\qquad d\mu_d=\pi^{-d}d^{2d}z,] where (d) is the physical mode number. Spectator vacuum modes do not change this probability. Exact reconstruction and its mechanism. The probabilities (M_2,\ldots,M_m), together with normalization, determine every mode weight, including zero and repeated weights. The new connection is from these optical probabilities to spectral power sums. A formal logarithm separates connected photon pairings; their strictly positive coefficients make the map triangular and invertible at every order. Standard Newton identities then recover the spectrum. For generic positive, distinct weights, the lower prefix through (M_{m-1}) leaves even the largest weight undetermined. A concrete pair of three-photon sources has spectra ((1/2,1/2,0)) and ((2/3,1/6,1/6)). Both give (M_2=11/64), while their three-copy probabilities are (155/6561) and (183/6561). The reconstructed largest weight also determines the coherent peak and full trace-norm distance to a pure coherent state: [q=\frac{N^Ne^{-N}}{N!}\max_jw_j,\qquad D_{\mathcal C}^{,2}=4(1-q).] Physical resource accounting. Highest interference order is distinct from total sample consumption. With (n_\ell) independent trials at order (\ell), the total number of input copies is (C_{\rm in}=\sum_{\ell=2}^{m}\ell n_\ell), and the incident photon count is (NC_{\rm in}). A finite two-hypothesis counting example is included. The exact inverse is not a uniform noisy-spectrum confidence theorem, and the order-necessity result concerns this specified vacuum-event hierarchy rather than arbitrary optical measurements. Mode-resolved one-particle measurements provide a different route to the spectrum. The integrated 13-page article contains two figure groups, complete proofs and references. It distinguishes the all-order optical-to-spectral connection from established replica-moment formulas and classical power-sum inversion.
When does finite phase-space entropy stop guaranteeing proximity to coherent light? This article determines sharp thresholds for two broad optical source classes when the number of modes may grow. It separates the roles of entropy, mode number and photon energy in approximating an optical state by a single pure coherent state. For an m-mode state ρ, write Qρ(α)=⟨α∣ρ∣α⟩, with dμm=π−md2mα, and define M(ρ)=2m∫Qρ(α)2dμm(α),s=−lnM(ρ),D(ρ)=infα∥ρ−∣α⟩⟨α∣∥1. Here s is the second-order Wehrl entropy excess and D is the full trace-norm distance, whose maximum is 2. The collision quantity M is also the output purity after half loss and can be accessed using interference of two independent, identically prepared copies. Exact source-class thresholds. The two classes are states diagonal in a common passive-mode number basis, and arbitrary states supported on the vacuum together with configurations whose photons occupy only one mode at a time in a common passive basis. The latter class retains arbitrary complex coherences, including vacuum coherences. For either class X, with FX(s)=supρ∈X,−lnM(ρ)=sD(ρ)2 and the supremum over all finite mode numbers, FX(s)=4⟺s≥ln4. The plateau is approached through growing-mode sequences, not attained by a finite-mode state. At its onset, vanishing coherent overlap and M→1/4 force the one-photon probability to approach one, without an energy-bound assumption. This probability rigidity does not assert convergence of the mean energy. Sharp boundary geometry and energy. For the full number-diagonal class D, the approach from below has the exact corner lims↑ln4ln4−s4−FD(s)=e4. A small vacuum component changes the optimal coherent peak and produces the sharp slope. Separate all-state energy bounds show that vanishing coherent overlap requires at least one photon asymptotically, or two photons for pure states. Explicit sources attain these limiting energy costs, and exact-entropy constructions include their additional photon costs. The proofs retain arbitrary photon tails and constants uniform in the mode number. The earliest-threshold and corner results apply to the source classes stated above. The integrated article contains the complete proofs and two figure groups.
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 Pe 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 (Pe+P−e)/2. For any normal two-arm state Ω with total mean photons at most E, and ∣e∣≤θ≤π/4, TrΩ(2Pe+P−e−P0)≤21+3Eθ2. The exact projector curvature on the total k-photon sector is k+3k2−2k. 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(ρi⊗σi): 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)=infα∥ρ−∣α⟩⟨α∣∥1,DU2=2(B−n)1∑i∈U[D(ρi)2+D(σi)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 DU2≤min{4,c4[δ+ην+ηn(B−n)Bln(1/βsig)]}. 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/Cˉ), where Cˉ≥C⋆ comes from the established sharp Wehrl theorem; the displayed examples use Cˉ=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.
How many uses of an optical device are needed to certify that it preserves quantum information? This article determines the optimal asymptotic number of coherent-light probes under an average photon budget. Matching upper and lower bounds show that a single optical mode carries an unavoidable logarithmic cost, while every fixed higher mode number follows a different power law. Certification task and resource accounting. The device is an arbitrary normal, memoryless bosonic quantum channel. The task distinguishes the identity from channels at least ε away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget Es. This guarantee includes inputs entangled with an unrestricted reference system. The physical probes are fresh coherent states with classically chosen amplitudes. Adaptive choices, retained quantum outputs and a final joint measurement are allowed. Every device call, including a vacuum pulse, is counted, and the actual illumination obeys ∑t=1TEK∥αt∥2≤TEp under every channel in the testing promise. The probe budget Ep is distinct from the signal budget Es defining the certification guarantee. Optimal mean-energy query laws. For fixed positive Es,Ep, fixed finite mode number m, and both testing errors at most δ, the minimax number of calls satisfies Tmean(1)(ε,δ)=ΘEs,Ep(ε−4logε1logδ1),Tmean(m)(ε,δ)=Θm,Es,Ep(ε−(2m+2)logδ1),m≥2. The bounds hold for sufficiently small ε, uniformly over 0<δ≤1/4. Independent probes drawn from a physical positive multiscale source, followed by individual binary target tests, attain these orders. The matching converse covers adaptive coherent-probe protocols with quantum output memory. Thus those additional processing capabilities do not improve the asymptotic order within the stated preparation model. The result specifies matching scaling laws, rather than exact finite-query counts or optimal numerical constants. Why one mode has an extra logarithm. Rare high-photon coherences can strongly affect an energy-bounded quantum input while leaving only a weak signature on coherent probes. The proof retains this interference through a positive coherent-frame Green kernel. Its spectral sum grows as tlogt for one mode and as tm for fixed m≥2. A complementary family of physical channels shows that the single-mode logarithm is necessary, rather than an artifact of the reconstruction bound. What source brightness changes. The optimal mean-only laws contrast with prescribed independent Gaussian modulation, for which logTG=Θ(ε−2) at fixed confidence. A fixed brightness cap gives logTcap=Θ(ε−2log(1/ε)). The article also establishes matching logarithmic orders under stretched-exponential brightness constraints and develops the conversion from average coherent-probe error to thermal Choi-state and energy-constrained worst-input guarantees. These are worst-case certification limits for the specified coherent-probe access model. The channel class retains infinite photon tails and imposes no output-energy bound. This updated, integrated 37-page article contains the main results, two figure groups, complete proofs and references.