Optimal measurements for quantum multi-parameter estimation with general states. (arXiv:1806.07337v2 [quant-ph] UPDATED)

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
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 guarantee the
existence of the optimal measurement, 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.

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