  An overwhelming majority of experiments in classical and quantum physics make
a priori assumptions about the dimension of the system under consideration.
However, would it be possible to assess the dimension of a completely unknown
system only from the results of measurements performed on it, without any extra
assumption? The concept of a dimension witness answers this question, as it
allows one to bound the dimension of an unknown classical or quantum system in
a device-independent manner, that is, only from the statistics of measurements
performed on it. Here, we report on the experimental demonstration of dimension
witnesses in a prepare and measure scenario. We use pairs of photons entangled
in both polarization and orbital angular momentum to generate ensembles of
classical and quantum states of dimensions up to 4. We then use a dimension
witness to certify their dimensionality as well as their quantum nature. Our
results open new avenues for the device-independent estimation of unknown
quantum systems and for applications in quantum information science.
