  Using Bloch's parametrization for qudits ($d$-level quantum systems), we
write the Hilbert-Schmidt distance (HSD) between two generic $n$-qudit states
as an Euclidean distance between two vectors of observables mean values in
$\mathbb{R}^{\Pi_{s=1}^{n}d_{s}^{2}-1}$, where $d_{s}$ is the dimension for
qudit $s$. Then, applying the generalized Gell Mann's matrices to generate
$SU(d_{s})$, we use that result to obtain the Hilbert-Schmidt quantum coherence
(HSC) of $n$-qudit systems. As examples, we consider in details one-qubit,
one-qutrit, two-qubit, and two copies of one-qubit states. In this last case,
the possibility for controlling local and non-local coherences by tuning local
populations is studied and the contrasting behaviors of HSC, $l_{1}$-norm
coherence, and relative entropy of coherence in this regard are noticed. We
also investigate the decoherent dynamics of these coherence functions under the
action of qutrit dephasing and dissipation channels. At last, we analyze the
non-monotonicity of HSD under tensor products and report the first instance of
a consequence (for coherence quantification) of this kind of property of a
quantum distance measure.
