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Abstract

We study estimation of degree-n normally ordered polynomial energies from N copies of a centered thermal Gaussian state with unknown occupation covariance A ⪰ aI, a > 0. A positive copy-averaged observable has an exact coefficient-dependent variance and minimizes variance at each covariance for every fixed polynomial among globally unbiased measurements and decisions. With γ = 1 + a⁻¹, its sharp worst-polynomial variance gives the arbitrary-bias minimax scale γn²/N. At fixed a and growing degree, the minimax relative squared risk tends exactly to 1 − e^(−λ) when this scale tends to λ ∈ (0, ∞). Standard heterodyne records have an explicit optimal unbiased estimator and the corresponding minimax scale γ²n²/N, yielding a matched low-brightness measurement separation. The proofs combine classical symmetry averaging and Gaussian moment methods with completeness of the copy-invariant thermal family and a supported-prior argument for negative-binomial photon data. Worked comparisons show how statistical pooling and polynomial coefficient geometry reduce fluctuations at a fixed copy budget.