Optimal unbiased Gaussian quantum estimation and the cost of unknown squeezing

Abstract

We determine minimum-variance unbiased estimators of scalar observables from N copies of an unknown Gaussian quantum state. Real orthogonal mixing of the copies defines an expectation-complete invariant model. Its completeness makes the averaged observable optimal at every centered Gaussian state, including squeezed, multimode and pure states, among all globally unbiased integrable quantum procedures. The theorem covers bounded copy observables and number-preserving polynomials, with the unbounded case justified by absolute first-moment forms and exponential photon-number domains. Polynomial energies admit an exact finite Wick formula. For the single-mode quartic energy, we derive the optimal unbiased observable and its variance, then prove a sharp resource consequence allowing arbitrary bias: with occupation at least a, both collective quantum measurements and standard heterodyne have minimax relative risk of order min{1, 1/(Na²)} uniformly for 0 < a ≤ 1/4. Thus admitting unknown squeezing changes the low-brightness quantum copy cost from inverse-linear to inverse-quadratic occupation, matching the heterodyne order. An explicit heterodyne pair statistic attains the upper order, while a squeezed-thermal testing experiment gives the quantum lower bound. Joint location-scatter completeness extends pointwise unbiased optimality to centered targets with unknown displacement using the residual copies.