  Analyzing the dimension of an unknown quantum system in a device-independent
manner, i.e., using only the measurement statistics, is a fundamental task in
quantum physics and quantum information theory. In this paper, we consider this
problem in the prepare-and-measure scenario. Specifically, we provide a lower
bound on the dimension of the prepared quantum systems which is a function that
only depends on the measurement statistics. Furthermore, we show that our bound
performs well on several examples. {In particular}, we show that our bound
provides new insights into the notion of dimension witness, and we also use it
to show that the sets of restricted-dimensional prepare-and-measure
correlations are not always convex.
