  The process in which a photon splits into three after the collision with a
free electron (triple Compton effect) is the most basic process for the
generation of a high-energy multi-particle entangled state composed out of
elementary quanta. The cross section of the process is evaluated in two
experimentally realizable situations, one employing gamma photons and
stationary electrons, and the other using keV photons and GeV electrons of an
x-ray free electron laser. For the first case, our calculation is in agreement
with the only available measurement of the differential cross section for the
process under study. Our estimates indicate that the process should be readily
measurable also in the second case. We quantify the polarization entanglement
in the final state by a recently proposed multi-particle entanglement measure.
