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We determine the ultimate potential of quantum imaging for boosting theresolution of a far-field, diffraction-limited, linear imaging device withinthe paraxial approximation. First we show that the problem of estimating theseparation between two point-like sources is equivalent to the estimation ofthe loss parameters of two lossy bosonic channels, i.e., the transmissivitiesof two beam splitters. Using this representation, we establish the ultimateprecision bound for resolving two point-like sources in an arbitrary quantumstate, with a simple formula for the specific case of two thermal sources. Wefind that the precision bound scales with the number of collected photonsaccording to the standard quantum limit. Then we determine the sources whoseseparation can be estimated optimally, finding that quantum-correlated sources(entangled or discordant) can be super-resolved at the sub-Rayleigh scale. Ourresults set the upper bounds on any present or future imaging technology, fromastronomical observation to microscopy, which is based on quantum detection aswell as source engineering.

Ultimate precision bound of quantum and sub-wavelength imaging