  Photons with a twisted phase front carry a quantized amount of orbital
angular momentum (OAM) and have become important in various fields of optics,
such as quantum and classical information science or optical tweezers. Because
no upper limit on the OAM content per photon is known, they are also
interesting systems to experimentally challenge quantum mechanical prediction
for high quantum numbers. Here, we take advantage of a recently developed
technique to imprint unprecedented high values of OAM, namely spiral phase
mirrors (SPM), to generate photons with more than 10,000 quanta of OAM.
Moreover, we demonstrate quantum entanglement between these large OAM quanta of
one photon and the polarization of its partner photon. To our knowledge, this
corresponds to entanglement with the largest quantum number that has been
demonstrated in an experiment. The results may also open novel ways to couple
single photons to massive objects, enhance angular resolution and highlight OAM
as a promising way to increase the information capacity of a single photon.
