  We establish conditions to characterize probability measures by their
$L^{p}$-quantization error functions in both $\mathbb{R}^{d}$ and Hilbert
settings. This characterization is two-fold: static (identity of two
distributions) and dynamic (convergence for the $L^p$-Wasserstein distance). We
first propose a criterion on the quantization level $N$, valid for any norm on
$\mathbb{R}^{d}$ and any order $p$ based on a geometrical approach involving
the Vorono\"i diagram. Then, we prove that in the $L^2$-case on a (separable)
Hilbert space, the condition on the level $N$ can be reduced to $N=2$, which is
optimal. More quantization based characterization cases on dimension 1 and a
discussion of the completeness of a distance defined by the quantization error
function can be found in the end of this paper.
