  The error in estimating the separation of a pair of incoherent sources from
radiation emitted by them and subsequently captured by an imager is
fundamentally bounded below by the inverse of the corresponding quantum Fisher
information (QFI) matrix. We calculate the QFI for estimating the full
three-dimensional (3D) pair separation vector, extending previous work on pair
separation in one and two dimensions. We also show that the pair-separation QFI
is, in fact, identical to source localization QFI, which underscores the
fundamental importance of photon-state localization in determining the ultimate
estimation-theoretic bound for both problems. We also propose general
coherent-projection bases that can attain the QFI in two special cases. We
present simulations of an approximate experimental realization of such quantum
limited pair superresolution using the Zernike basis, confirming the
achievability of the QFI bounds.
