Choosing optical inputs under Gaussian noise: non-Gaussian advantage and competing optima

An optical input that best resists Gaussian noise need not be Gaussian, and its optimal form can change with the output objective. We establish both effects for a single-mode additive Gaussian channel with unequal quadrature noise and a fixed mean photon budget. Below the noise-matching energy E*, physical states with nonnegative Wigner functions exceed the output purity of every Gaussian mixture at the same energy. An explicit source retains an advantage greater than 1.4 × 10−4 over a finite range of channel and preparation errors. As Δ = E* − E ↓ 0, the optimal gain above the Gaussian benchmark is CpΔ4 + o(Δ4) for each fixed output moment Tr[(Φρ)p], p > 1, and for the largest output eigenvalue. A four-photon amplitude θpΔ2 attains this gain, with different objectives selecting different amplitudes. Consequently, no attainable output spectrum majorizes all others near matching. For a fixed finite catalogue of objectives, the largest common fraction of their separately optimal leading gains is 4θ−θ+/(θ− + θ+)2, attained by one Wigner-positive preparation; θ− and θ+ are the extreme preferred amplitudes. These results identify when to extend an optical input search beyond Gaussian sources and how to choose a single preparation when output objectives compete.