Sharp estimation costs for thermal Gaussian energies with known background

Abstract

We determine the sampling cost of estimating high-order source energies from thermal Gaussian states with a known additive background. The source has unknown occupation covariance above a positive spectral floor, and the known background covariance may fail to commute with it. A finite inverse-heat correction produces a signed observable whose exact coefficient-dependent variance is the minimum among all globally unbiased quantum procedures. Its sharp worst-polynomial envelope gives matching arbitrary-bias minimax bounds. As degree grows, the quantum and standard-heterodyne risks converge to one minus an exponential at their respective resource scales, uniformly along arbitrary known sequences of source floors, attenuation, backgrounds and finite dimensions. The lower proof uses a normalized scalar likelihood whose relative curvature remains controlled through vanishing and diverging occupation regimes. The exact ratio of the two sample-cost coefficients is one plus the inverse output occupation floor. Consequently quantum relative consistency together with heterodyne complete collapse is possible precisely when that floor tends to zero. Known-background subtraction corrects expectations while retaining a quantified fluctuation cost. Scalar and noncommuting two-mode examples illustrate the exact risks and measurement budgets.