General tradeoff relation of fundamental quantum limits for linear multiparameter estimation

Abstract: Linear measurement is an important class of measurements for sensing classical signals including gravitational wave (GW), dark matter, infrared ray, rotation rate, etc. In this Letter, we focus on multiparameter linear measurement and establish a general tradeoff relation that tightly constrains the fundamental quantum limits of two independent parameters in a monochromatic classical signal detected by any linear quantum device. Such a tradeoff relation is universal and fundamental for multiparameter linear measurement since arising from Heisenberg's uncertainty principle. Compared with the Holevo Cram\'er-Rao bound, our tradeoff bound can completely identify the dependence between the attainable precision limits on estimated parameters. The dependence becomes more obvious such that the individual precision can not simultaneously reach the quantum limit as the so-called incompatible coefficient rises. Eventually, we find a necessary condition under which an optimal measurement protocol can saturate the general tradeoff relation, and show that the measurement phase can be tuned for adjusting different precision weight. This result is related to many applications, particularly detuned GW sensors for searching post-merger remnants due to the direct relation between the detuned frequency and incompatible coefficient.