Ultimate Precision Bound of Quantum and Subwavelength Imaging

We determine the ultimate potential of quantum imaging for boosting the resolution of a far-field, diffraction-limited, linear imaging device within the paraxial approximation. First, we show that the problem of estimating the separation between two pointlike sources is equivalent to the estimation of the loss parameters of two lossy bosonic channels, i.e., the transmissivities of two beam splitters. Using this representation, we establish the ultimate precision bound for resolving two pointlike sources in an arbitrary quantum state, with a simple formula for the specific case of two thermal sources. We find that the precision bound scales with the number of collected photons according to the standard quantum limit. Then, we determine the sources whose separation can be estimated optimally, finding that quantum-correlated sources (entangled or discordant) can be superresolved at the sub-Rayleigh scale. Our results apply to a variety of imaging setups, from astronomical observation to microscopy, exploiting quantum detection as well as source engineering.