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. Quantum estimation with continuous variable systems faces challenges in deriving accessible and practical results for practitioners. Here, the authors present a unified framework for computing multiparameter bounds for Gaussian states, demonstrating their approach through semidefinite programming, which could significantly enhance precision in quantum measurements.

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A unified framework is introduced for computing the Holevo Cramér-Rao bound (HCRB) for arbitrary multimode Gaussian states in multiparameter quantum metrology. By reformulating the HCRB as a semidefinite program (SDP) that depends only on first and second moments of the state and their parametric derivatives, the approach avoids the intractable optimization over Hermitian operators in infinite-dimensional systems. The same phase-space formulation also yields SDP forms for the symmetric logarithmic derivative (SLD) and right logarithmic derivative (RLD) bounds, plus analytical results for two-parameter single-mode covariance matrix estimation. The method is demonstrated on simultaneous estimation of phase and loss, and joint estimation of displacement and squeezing. Python code via CVXPY is available on GitHub.

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