Optimal measurements for quantum multiparameter estimation with general states

We generalize the approach by Braunstein and Caves [Phys. Rev. Lett. 72, 3439 (1994)] to quantum multi-parameter estimation with general states. We derive a matrix bound of the classical Fisher information matrix due to each measurement operator. The saturation of all these bounds results in the saturation of the matrix Helstrom Cram\'er-Rao bound. Remarkably, the saturation of the matrix bound is equivalent to the saturation of the scalar bound with respect to any given positive definite weight matrix. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the Helstrom Cram\'er-Rao bound associated with a general quantum state. To saturate the Helstrom bound with separable measurements or collective measurement entangling only a small number of identical states, we find it is necessary for the symmetric logarithmic derivatives to commute on the support of the state. As an important application of our results, we construct several local optimal measurements for the problem of estimating the three-dimensional separation of two incoherent optical point sources.