A classical current in a conductor radiates a classical electromagnetic field. We explore some properties of the field radiated by a conductor when electron transport must be described by quantum mechanics, i.e. when the electron current becomes quantum itself. We have measured the quadratures of the field generated by a tunnel junction placed at ultra-low temperature in the presence of ac+dc voltage bias. We demonstrate the existence of squeezing as well as entanglement between quadratures at two different frequencies, thus proving that the electron shot noise generates a quantum electromagnetic field. We also discuss the link between photon statistics of the radiated field, deduced form measurements with linear detectors, and some correlators of the electron current.