Quantum limits of parameter estimation in long-baseline imaging
Distributed aperture telescopes are a well-established approach for boosting resolution in astronomical imaging. However, theoretical limits on quantitative imaging precision, and the fundamentally best possible beam-combining and detection schemes to use with such arrays, remain largely unexplored. Using mathematical tools of the quantum and classical Cramer-Rao bounds, we perform analyses showing the fundamental origins of the enhancement provided by distributed imaging systems, over and above a single monolithic telescope, and consider the precision with which one can estimate any desired parameter embedded in a scene's incoherent radiation with a multi-aperture imaging system. We show how quantum-optimal measurements can be realized via beam-combination strategies of two classes: (1) multi-axial: where light from different apertures is directed to a common focal plane, e.g., of a segmented-aperture telescope; and (2) co-axial: where light collected at each aperture, e.g., telescope sites of a long-baseline array, is routed to an optical interferometer. As an example, we show an explicit calculation of the quantum Fisher information (QFI) for estimating the angular separation between two-point emitters using two identical apertures separated by a baseline distance. We show that this QFI splits instructively into additive contributions from the single apertures and from the baseline. We quantify the relative benefits of intra-telescope (e.g., spatial-mode) optical processing and inter-telescope beam combination. We show how both receiver designs can capture both sources of information and discuss how similar methods could be extended to more general imaging tasks. We discuss translating QFI-attaining measurements to explicit receiver designs, and the use of pre-shared entanglement to achieve the QFI when it is impractical to co-locate and combine light collected by the apertures.