Ultimate precision of joint quadrature parameter estimation with a Gaussian probe

The Holevo Cramér-Rao bound is a lower bound on the sum of the mean-square error of estimates for parameters of a state. We provide a method for calculating the Holevo Cramér-Rao bound for estimation of quadrature mean parameters of a Gaussian state by formulating the problem as a semidefinite program. In this case, the bound is tight; it is attained by purely Gaussian measurements. We consider the example of a symmetric two-mode squeezed thermal state undergoing an unknown displacement on one mode. We calculate the Holevo Cramér-Rao bound for joint estimation of the conjugate parameters for this displacement. The optimal measurement is different depending on whether the state is entangled or separable.