Optimal measurements in simultaneous multi-parameter estimation of quantum systems

Simultaneous estimation of multiple parameters is a well-known challenge in quantum metrology. The underlying difficulty is to identify the common optimal measurements for all the parameters, which typically do not coincide or commute. For a general probe state and a projective measurement of arbitrary rank, we find the necessary and sufficient conditions under which the measurement gives rise to the multi-parameter quantum Cramer-Rao matrix bound. We also give an application of these conditions to the specific problem of estimating three-dimensional separation of two point incoherent sources of equal intensities from single photon measurements. By considering the hard-aperture and paraxial approximated pupil function, we find a local optimal measurement for simultaneous estimation of the small three dimensional separation. Furthermore, regardless of the magnitude of longitudinal separation, a local optimal measurement for simultaneously estimating the small transverse separation is also found. The saturation conditions of the multi-parameter quantum Cramer-Rao bound may be further explored in quantum sensing and imaging.