  Einstein, Podolsky and Rosen's prediction on incompleteness of quantum
mechanics was overturned by experimental tests on Bell's inequality that
confirmed the existence of quantum entanglement. In X-ray optics, entangled
photon pairs can be generated by X-ray parametric down conversion (XPDC), which
has certain wavelength window. Meanwhile, free electron laser (FEL) has
successfully lased at X-ray frequencies recently. However, FEL is usually seen
as a classical light source, and its quantum effects are considered minor
corrections to the classical theory. Here we investigate entangled X-ray photon
pair emissions in FEL. We establish a theory for coherently amplified entangled
photon pair emission from microbunched electron pulses in the undulator. We
numerically demonstrate the properties of entangled emission, and provide a
scheme to generate highly entangled X-ray photon pairs, which is of great
importance in X-ray quantum optics. Our work shows a unique advantage of FELs
over synchrotrons in entangled X-ray photon pair generation.
