LAUSR

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 projector of a measurement. The saturation of all these bounds results in the saturation of the multi-parameter quantum Cram\'er-Rao bound. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the multi- parameter quantum Cram\'er-Rao bound associated with a general quantum state. We find that nonlocal measurements on replicas of a pure or full ranked state do not help saturate the multi- parameter quantum Cram\'er-Rao bound if no optimal measurement exists for a single copy of such a 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.