  We analyze the transfer of quantum information to detected classical
information in four-dimensional scanning transmission electron microscopy. In
estimating the moduli and phases of the Fourier coefficients of the sample's
electrostatic potential, we find that near-optimum information transfer is
achieved by a delocalized speckled probe, which attains about half of the
available quantum Fisher information. The quantum limit itself is precluded due
to detecting the scattering in momentum space. We compare with direct
phase-contrast imaging, where a Zernike phase condition attains the quantum
limit for all spatial frequencies admitted by the optical system. Our
conclusions also apply to other forms of coherent scalar radiation, such as
visible light and x-rays.
