Closed-form Bayesian quantum estimation of Gaussian states

Bayesian quantum estimation provides a robust framework for quantum technologies, especially in scenarios with limited data and minimal prior information. Yet, its application to continuous-variable Gaussian systems has remained limited and largely numerical due to the complexity of the underlying parameter integrals. Here, we introduce a variational framework reducing the optimisation over measurements and estimators to a finite-dimensional linear problem and admitting closed-form solutions. This is achieved by restricting the analysis to operators polynomial in the canonical quadratures, leading to solutions with a geometric interpretation as orthogonal projections of the global optimum. We further derive a necessary and sufficient condition for global optimality. Through single-shot examples, we show that the framework yields experimentally feasible strategies based on Gaussian operations and quadrature measurements that are either optimal or near-optimal, and that replacing the induced estimator with the posterior mean further improves performance towards the global optimum.