Sufficient support size of measurements for quantum estimation
In quantum estimation for a $d$-parameter family of density operators on a finite-dimensional Hilbert space $\mathcal{H}$, an estimator is specified by a pair $\left(M,\hatθ\right)$, where $M$ is a POVM with a finite outcome set $Ω$ and $\hatθ:Ω\to\mathbb{R}^{d}$ is a classical estimator map. Since the number of outcomes $\left|Ω\right|$ is a priori unbounded, the space of admissible POVMs is vast, which makes the search for optimal estimators difficult. In this paper, for the minimization of the weighted trace of the mean squared error among locally unbiased estimators, we prove that it suffices to consider POVMs with at most $\left(\dim\mathcal{H}\right)^{2}+d(d+1)/2-1$ outcomes, and that the optimization can be restricted to rank-one measurements. For Bayesian estimation with a general loss function, we show that rank-one POVMs with at most $(\dim\mathcal{H})^{2}$ outcomes are sufficient for the infimum value of the Bayes risk. Furthermore, when the model admits a real sufficient subalgebra, we show that the $\left(\dim\mathcal{H}\right)^{2}$ term in the above support-size bounds can be reduced in both the locally unbiased and Bayesian settings. These bounds substantially reduce the search space for optimal measurements and justify restricting numerical optimization to rank-one POVMs with finitely many outcomes.