Abstract
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.
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The data supporting the findings of this study are available from the first author upon request.
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The theoretical results are reproducible from the analytical formulas and derivations presented in the manuscript. A Jupyter notebook containing the Python code used to generate the figures is available on Github91.
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Acknowledgements
F. A. thanks Animesh Datta and Dominic Branford for fruitful discussions in the early stages of this project. M.G.G. and F.A. thank Matteo Paris and Alessio Serafini for several discussions on quantum estimation theory and Gaussian quantum states. S.C. acknowledges support by the China Scholarship Council. M.G.G. acknowledges support from MUR and Next Generation EU via the NQSTI-Spoke2-BaC project QMORE (contract no. PE00000023-QMORE). F.A. acknowledges financial support from Marie Skłodowska-Curie Action EUHORIZON-MSCA-2021PF-01 (project QECANM, grant no. 101068347).
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F.A. conceived the project and obtained preliminary results. S.C. performed analytical and numerical calculations. M.G.G. checked and streamlined derivations and calculations. F.A. and M.G.G. jointly supervised the project. All authors discussed the results and wrote the paper.
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Shoukang, C., Genoni, M.G. & Albarelli, F. Efficiently evaluating Holevo, RLD and SLD Cramér-Rao bounds for multiparameter quantum estimation with Gaussian states. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02550-6
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DOI: https://doi.org/10.1038/s42005-026-02550-6
