Efficiently evaluating Holevo, RLD and SLD Cramér-Rao bounds for multiparameter quantum estimation with Gaussian states

Continuous-variable Gaussian states are ubiquitous in quantum science, describing relevant regimes in optics, optomechanics, and atomic ensembles. In multiparameter quantum metrology, their ultimate precision limit is set by the Holevo Cramér-Rao bound (HCRB), which accounts for measurement incompatibility. However, evaluating the HCRB in infinite-dimensional systems is challenging due to the required optimization over Hermitian operators. Here we introduce an efficient, general method to compute the HCRB for arbitrary multimode Gaussian states by reformulating it as a semidefinite program (SDP) depending only on the first and second moments of the state and their parametric derivatives. This phase-space formulation shows that observables up to quadratic order in the canonical operators suffice to evaluate the bound. The same framework yields SDP forms of the symmetric and right logarithmic derivative (SLD and RLD) bounds and analytical results for two parameters encoded in a single-mode covariance matrix. We demonstrate the approach in two scenarios where both first and second moments vary with the parameters: simultaneous estimation of phase and loss, and joint estimation of displacement and squeezing. Our results provide conceptual insight into multiparameter estimation with Gaussian states and enable practical applications of the HCRB.