Quantum and classical Fisher information in four-dimensional scanning transmission electron microscopy

We analyze the quantum limit of sensitivity in four-dimensional scanning transmission electron microscopy (4D-STEM), which has emerged as a favored technique for imaging the structure of a wide variety of materials, including biological and other radiation-sensitive materials. 4D-STEM is an indirect (computational) imaging technique, which uses a scanning beam, and records the scattering distribution in momentum (diffraction) space for each beam position. We find that, in measuring a sample's electrostatic potential, the quantum Fisher information from 4D-STEM can match that from real-space phase-contrast imaging. Near-optimum quantum Fisher information is achieved using a delocalized speckled probe. However, owing to the detection in the diffraction plane, 4D-STEM ultimately enables only about half of the quantum limit, whereas Zernike phase-contrast imaging enables the quantum limit for all spatial frequencies admitted by the optical system. On the other hand, 4D-STEM can yield information on spatial frequencies well beyond those accessible by phase-contrast TEM. Our conclusions extend to analogous imaging modalities using coherent scalar visible light and x-rays.