Degree-dependent minimax laws for Gaussian factorial moments

Abstract

We classify the copy budgets for estimating photon-number factorial moments of an unknown centered single-mode Gaussian state. Both occupation and complex anomalous covariance are unknown, and arbitrary bias is allowed. For every degree n ≥ 2, collective quantum procedures and standard heterodyne have minimax squared relative risk of order min{1, (1 + a⁻¹)²n²/N}, uniformly in degree, copy count and positive occupation floor a. Degree one is exceptional: its quantum scale is (1 + a⁻¹)/N. We prove complete endpoint classifications: risk tends to zero exactly when its resource scale vanishes, and for n ≥ 2 tends to one exactly when that scale diverges. Bounded positive scales keep both risks away from zero and one. Nevertheless one scalar resource scale cannot determine all critical risks. A growing-degree boundary experiment has a Poisson observation with a modified-Bessel target; its exact classical minimax risk yields an explicit quantum lower. At the same occupation floor and exactly the same resource scale, this lower separates from a published quartic upper. The universal copy-budget order thus coexists with genuinely degree-dependent critical behavior.