  Device independent dimension witnesses provide a lower bound on the
dimensionality of classical and quantum systems in a "black box" scenario where
only correlations between preparations, measurements and outcomes are
considered. We address the problem of the robustness of dimension witnesses,
namely that to witness the dimension of a system or to discriminate between its
quantum or classical nature, even in the presence of loss. We consider the case
when shared randomness is allowed between preparations and measurements and we
provide a threshold in the detection efficiency such that dimension witnessing
can still be performed.
