  A number of visions for a new generation of dose-efficient electron
microscopes have been advanced. These proposals, while inspired by quantum
principles, make little contact with the broader field of quantum metrology. We
discuss a framework calculating the amount of information carried by each
electron. This makes it possible to evaluate the potential effectiveness of any
particular microscope architecture relative to the quantum limit for
information per dose. In the case of phase imaging, we argue this limit is at
least an order of magnitude beyond what is possible with aberration-free
Zernike phase contrast.
