Transmission electron microscopy at the quantum limit

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 for 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.