Exact low-brightness minimax risk for quartic Gaussian energy

Abstract

We determine the exact critical minimax relative risk for estimating the normally ordered quartic energy of an unknown centered single-mode Gaussian state. The occupation and complex anomalous covariance are both unknown, and arbitrary biased collective quantum procedures are allowed. If the occupation floor a tends to zero and the copy budget satisfies Na² → λ ∈ (0, ∞), the minimax risk tends to (1 + 2√λ)⁻². A vanishing fraction of photon-counting observations estimates occupation; an affine correction of a known unbiased collective observable attains the limit uniformly over the unbounded state family. For the lower bound, a harmonic decomposition and common-phase averaging reduce a compact boundary experiment to a Poisson count with a positive shifted target. A proper Bayes equalizer solves that decision problem. We also prove complete collapse when Na² → 0 and sharp high-information limits: the quantum risk is asymptotic to 1/(4Na²), whereas standard heterodyne has risk 4/(Na²). Thus the previously known equality of copy-cost orders coexists with a sharp factor-sixteen asymptotic efficiency advantage.