Ultimate tradeoff relation of quantum precision limits in multiparameter linear measurement

Linear measurements are widely applied in sensing classical signals, e.g., gravitational wave (GW), and are developing toward joint measurement of multiple parameters. In this work, focusing on multiparameter linear measurements of classical monochromatic signals, we establish an inherent tradeoff relation that tightly constrains the quantum limits on estimation precision. The tradeoff relation is fundamental since it is rooted in Heisenberg's uncertainty principle, and fully characterizes the dependence between the attainable precision limits on the estimated parameters. Eventually, we identify a necessary condition under which an optimal measurement protocol saturates the tradeoff relation, and show that the measurement phase can be regulated to implement flexible allocation of precision weights. Our finding can offer valuable guidance for detuned GW sensors in ultra-sensitive searches for post-merger remnants.