A certified non-Gaussian encoding advantage in single-photon phase-space detection

For a fixed optical receiver, optimizing the classical modulation of Gaussian signals need not reach the best communication rate. This work gives a rigorous way to certify the advantage of a non-Gaussian signal constellation against every one-mode Gaussian alphabet, and to quantify how much of that advantage survives specified receiver and preparation errors. What is compared The receiver is the displacement-covariant phase-space measurement generated by a single-photon ancillary state. It is not a binary photon-click detector. The cost is the actual mean transmitted photon number E; the receiver is fixed, and ancillary preparation and local-oscillator overhead are not included in that signal budget. Let C(E) be the unrestricted capacity and CG(E) the optimum over Gaussian letters with arbitrary Borel modulation, displacement, squeezing and mixedness. For the ideal receiver, CG(E) < C(E), for every 0 < E < ∞. The comparison is global: it does not restrict the Gaussian competitor to a finite constellation or a locally optimized ansatz. A finite-energy certificate with a calibration interval For the actual unflagged receiver seed σv = (1 − v)|1⟩⟨1| + v|0⟩⟨0|, the paper constructs four equiprobable phase-rotated signals supported on photon numbers 0–3. Each signal has exactly 0.1 mean photons. Write I4,v for their mutual information and CG(0.1,v) for the Gaussian optimum of that same receiver. At v = 0.01, in nats per use, I4,0.01 > 0.0839244, CG(0.1,0.01) < 0.0839162,so the certified excess is greater than 8.2 × 10−6. The same constellation obeys I4,v − CG(0.1,v) > 5 × 10−6 for every 0.009 ≤ v ≤ 0.011. This is a continuum guarantee, obtained from a supporting line for the same-source information and a convex upper chord for the Gaussian optimum. It is not interpolation between sampled advantages. The proof retains the full photon tail of Gaussian competitors. Phase tolerance at unchanged energy Suppose all four signals share static photon-number phases θn while retaining their exact amplitudes. After removing irrelevant constant and linear phases, the two residual errors are ε2 = θ2 − 2θ1 + θ0 and ε3 = θ3 − 3θ1 + 2θ0, modulo 2π. If both residual magnitudes are at most δ, their information loss is at most Kδ², with K < 0.132762313267. Thus δ = 0.004 radians (about 0.229°) leaves a certified excess above 2.8 × 10−6 nat/use throughout the receiver interval, with photon cost still exactly 0.1. This phase guarantee concerns a common static source error. A separate trace-distance preparation bound requires an independent check of actual mean energy. Independent phase jitter and arbitrary detector noise are different error models. Collective encodings and interpretation For the ideal receiver, every fixed finite entangled Gaussian block retains a total deficit, and uniformly bounded common blocks retain a per-use deficit. Exact correlation and covariance-repair estimates identify necessary extensive resources for a Gaussian sequence to approach capacity. These results do not settle arbitrary growing-block Gaussian encodings. The quantitative advantage is small—about 0.01% of the rate at the certified point. Its significance is a robust, globally certified separation with explicit resource and error accounting, rather than a large throughput improvement or an experimental demonstration. The article contains the complete proofs, figures, exact reference data and error bounds.