Sharp calibration costs for high-order thermal Gaussian energies

Abstract

We determine the joint cost of source measurements and loss calibration when estimating high-order energies of thermal Gaussian states. The source has unknown occupation covariance bounded below by a positive constant, and a common unknown transmissivity lies in a fixed interior interval. We observe independent source copies and single-photon calibration trials. As the polynomial degree grows with the square roots of both sample sizes, the worst-polynomial relative minimax risk converges to one minus the exponential of a sum of source and calibration costs. The result allows arbitrary biased decisions and collective source measurements, and is uniform in mode dimension and polynomial coefficients. Finite-sample bounds also identify the source and calibration budgets uniformly over the occupation floor, with a matched comparison to standard heterodyne. A reciprocal-binomial correction gives an attaining estimator whose moments remain controlled even at zero calibration counts. A supported product prior gives the matching quantum lower bound. The exact law yields the optimal deterministic allocation under specified sample costs. We also determine the complementary limit when calibration is supplied only as an interval: finite-copy uncertainty combines with an exact identification floor through a hyperbolic-secant factor. These results quantify the calibration precision needed to estimate growing-order source energies.