Certifying quantum non-Gaussianity with an unknown common phase

An unknown optical phase can hide quantum non-Gaussianity in every individual mode while leaving it certifiable in the joint light. This paper shows when Gaussian measurements can exclude the complete Gaussian convex hull without knowing the source's common phase, and gives explicit finite-sample tests. The results guide whether to preserve phase relations between emissions and how to process the resulting measurement records. 1. Certification against the full Gaussian model. Let \(\mathcal G_n\) be the trace-norm-closed convex hull of all physical \(n\)-mode Gaussian states. For a fixed known \(\rho\notin\mathcal G_1\), form the common-phase state \[\Omega_n=\int_0^{2\pi}(U_\phi\rho U_\phi^\dagger)^{\otimes n}\,\frac{d\phi}{2\pi},\qquad U_\phi=e^{-i\phi N}.\] Finite Gaussian measurements with a laboratory phase reference and bounded classical scores achieve arbitrarily small false-positive and missed-detection probabilities. The null includes arbitrary Gaussian entanglement and classical correlations, with unrestricted displacement, squeezing and energy. In particular, \[\inf_{\sigma_n\in\mathcal G_n}\|\Omega_n-\sigma_n\|_1\longrightarrow2.\] The norm is the full, unhalved trace norm. An unavailable common phase therefore differs operationally from independently randomising the phase of each emission. 2. An exact coherence boundary and a joint-only resource. For the displaced two-Gaussian-packet family, centred by its actual mean before pure loss, the Gaussian-hull boundary is \[\rho_{\eta,\nu}\in\mathcal G_1\quad\Longleftrightarrow\quad\nu\le\nu_c,\qquad \nu_c=\exp\!\left[-\frac{v^2R}{2(1-\eta)(v-1/2)^2}\right].\] Here \(v>1/2\) is the packet quadrature variance, \(\sqrt R\) the momentum displacement, \(0<\tau,\eta<1\) the packet weight and transmissivity, and \(0\le\nu\le1\) the coherence visibility. The same membership boundary holds after common-phase averaging for every \(n\ge3\). At \[(v,R,\tau,\nu,\eta)=(20,2/9,9/50,1,1/2),\] each phase-averaged single mode is a Gaussian mixture, while the joint common-phase state lies outside \(\mathcal G_n\) for \(n\ge3\). Independent phase averaging instead gives a Gaussian mixture at every block size. These states are separable and Wigner-positive: their difference is carried by phase relations across emissions. 3. The same two-packet source has an explicit finite certificate. Heterodyne phase training, a finite homodyne score and an eight-direction bounded tail test provide a complete certificate for the preceding parameter choice. With ideal Gaussian readout, the sufficient raw acquisition budget and certified errors are \[n_{\rm raw}=10^{15}+2(4\times10^{15})=9\times10^{15},\qquad P_{\rm FP}<3.86\times10^{-22},\qquad P_{\rm FN}<0.002068.\] This conservative budget establishes finite-sample certification of the joint-only resource; it is not an optimised experimental acquisition requirement. The PDF contains the measurement coefficients and complete proof. 4. A complementary two-million-trial implementation. For a displaced lossy single-photon source with unknown common phase, the target conditional state is \[\omega_\varphi=D(e^{i\varphi})\frac{|0\rangle\langle0|+|1\rangle\langle1|}{2}D^\dagger(e^{i\varphi}).\] A bounded homodyne score uses the full-Gaussian one-photon benchmark \(3\sqrt3/(4e)\), 128 phase candidates and decision threshold 0.490. Including the specified source and readout tolerances, \[n=2\times10^6,\qquad P_{\rm FP}<1.86\times10^{-4},\qquad P_{\rm FN}<7.908\times10^{-3}.\] All vacuum emissions, loss events and measurement outcomes count towards the raw budget. This is a separate source family from the two-packet example. Together, the results distinguish unknown common phase from destructive phase randomisation and connect full-model exclusion to explicit Gaussian-measurement scores, error bounds and acquisition costs. The 15-page PDF includes the manuscript, two figures and complete supplementary proofs.