  How does the quantum-to-classical transition of measurement occur? This
question is vital for both foundations and applications of quantum mechanics.
Here, we develop a new measurement-based framework for characterizing the
classical and quantum free electron-photon interactions and then experimentally
test it. We first analyze the transition from projective to weak measurement in
generic light-matter interactions and show that any classical
electron-laser-beam interaction can be represented as an outcome of a weak
measurement. In particular, the appearance of classical point-particle
acceleration is an example of an amplified weak value resulting from weak
measurement. A universal factor quantifies the measurement regimes and their
transition from quantum to classical, where Gamma corresponds to the ratio
between the electron wavepacket size and the optical wavelength. This
measurement-based formulation is experimentally verified in both limits of
photon-induced near-field electron microscopy and the classical acceleration
regime using a dielectric laser accelerator. Our results shed new light on the
transition from quantum to classical electrodynamics, enabling to employ the
essence of wave-particle duality of both light and electrons in quantum
measurement for exploring and applying many quantum and classical light-matter
interactions.
