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Abstract

We determine the copy-budget hierarchy and an exact critical risk for estimating photon-number factorial moments of an unknown centered single-mode Gaussian state. Occupation and complex anomalous covariance are unknown, and bias is unrestricted. If the occupation floor a tends to zero, the degree n tends to infinity and Na²/n² → λ ∈ (0, ∞), the collective quantum minimax squared relative risk converges to 1 − exp{−1/(2λ)}. A global bound for a shrunk orthogonal-copy average attains the Poisson–Bessel lower bound. The upper follows from a physical rank-one majorant of the ordered Wick cross-contractions and a degree comparison for Gaussian moments. For every n ≥ 2, quantum procedures and standard heterodyne have risk of order min{1, (1 + a⁻¹)²n²/N}, uniformly in degree, copy count and positive floor. Risk tends to zero exactly when this scale vanishes and tends to one exactly when it diverges. Degree one retains a smaller quantum scale. At identical floors and resource scales, a quartic upper and the exact growing-degree limit differ. Thus the universal copy-budget order coexists with degree-dependent critical behavior.