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The Holevo Cram\'er Rao bound is a lower bound on the sum of the mean squareerror of estimates for parameters of a state. We provide a method forcalculating the Holevo Cram\'er-Rao bound for estimation of quadrature meanparameters of a Gaussian state by formulating the problem as a semidefiniteprogram. In this case, the bound is tight; it is attained by purely Guassianmeasurements. We consider the example of a symmetric two-mode squeezed thermalstate undergoing an unknown displacement on one mode. We calculate the HolevoCram\'er-Rao bound for joint estimation of the conjugate parameters for thisdisplacement. The optimal measurement is different depending on whether thestate is entangled or separable.

Ultimate precision of joint quadrature parameter estimation with a Gaussian probe