  A dimension witness is a criterion that sets a lower bound on the dimension
needed to reproduce the observed data. Three types of dimension witnesses can
be found in the literature: device-dependent ones, in which the bound is
obtained assuming some knowledge on the state and the measurements;
device-independent prepare-and-measure ones, that can be applied to any system
including classical ones; and device-independent Bell-based ones, that certify
the minimal dimension of some entangled systems. Here we consider the
Collins-Gisin-Linden-Massar-Popescu (CGLMP) Bell-type inequality for four
outcomes. We show that a sufficiently high violation of this inequality
witnesses $d \geq 4$ and present a proof-of-principle experimental observation
of such a violation. This presents a first experimental violation of the third
type of dimension witness beyond qutrits.
