  An important problem in quantum information processing is the certification
of the dimension of quantum systems without making assumptions about the
devices used to prepare and measure them, that is, in a device-independent
manner. A crucial question is whether such certification is experimentally
feasible for high-dimensional quantum systems. Here we experimentally witness
in a device-independent manner the generation of six-dimensional quantum
systems encoded in the orbital angular momentum of single photons and show that
the same method can be scaled, at least, up to dimension 13.
