Optimal quantum estimation of the displacement in optical imaging

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We present a general method to find the ultimate limits on the resolution of optical image on a surface perpendicular to the light beam propagation. We prove that the optimal sensitivity always obeys the standard quantum limit for a single mode light. In the absence of photon losses, we find the possibility to achieve the Heisenberg limit by using two-mode or multi-mode lights. We also use the variational approach to obtain the lower bound on the sensitivity considering the effects of photon losses.