  An accessible source for the production of entangled x-rays is crucial for
the field of high-energy quantum optics. Here, we present a detailed analysis
of the entanglement and polarisation of the two photons emitted by an electron
in an intense laser wave (nonlinear double Compton scattering), by working
within the framework of strong-field QED. By identifying a contribution to the
emission probability stemming from the electron being on-shell or off-shell
between the two photons emissions, we show that the entangled photons are
generated via the off-shell contribution, which can be distinguished from those
emitted via the on-shell channel by a polarisation measurement. We also provide
an intuitive picture to explain the entanglement and propose an experiment to
produce and isolate entangled x-rays.
