Multiparameter quantum estimation with Gaussian states: efficiently evaluating Holevo, RLD and SLD Cram\'er-Rao bounds

Multiparameter quantum estimation theory is crucial for many applications involving infinite-dimensional Gaussian quantum systems, since they can describe many physical platforms, e.g., quantum optical and optomechanical systems and atomic ensembles. In the multiparameter setting, the most fundamental estimation error (quantified by the trace of the estimator covariance matrix) is given by the Holevo Cram\'er-Rao bound (HCRB), which takes into account the asymptotic detrimental impact of measurement incompatibility on the simultaneous estimation of parameters encoded in a quantum state. However, the difficulty of evaluating the HCRB for infinite-dimensional systems weakens the practicality of applying this tool in realistic scenarios. In this paper, we introduce an efficient numerical method to evaluate the HCRB for general Gaussian states, by solving a semidefinite program involving only the covariance matrix and first moment vector and their parametric derivatives. This approach follows similar techniques developed for finite-dimensional systems, and hinges on a phase-space evaluation of inner products between observables that are at most quadratic in the canonical bosonic operators. From this vantage point, we can also understand symmetric and right logarithmic derivative scalar Cram\'er-Rao bounds under the same common framework, showing how they can similarly be evaluated as semidefinite programs. To exemplify the relevance and applicability of this methodology, we consider two paradigmatic applications, where the parameter dependence appears both in the first moments and in the covariance matrix of Gaussian states: estimation of phase and loss, and estimation of squeezing and displacement.