Mode-spectrum reconstruction from multicopy optical interference

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.