  We study an analogous Bloch sphere representation of higher-level quantum
systems using the Heisenberg-Weyl operator basis. We introduce a
parametrization method that will allow us to identify a real-valued Bloch
vector for an arbitrary density operator. Before going into arbitrary $d$-level
($d\geq 3$) quantum systems (qudits), we start our analysis with three-level
ones (qutrits). It is well known that we need at least eight real parameters in
the Bloch vector to describe arbitrary three-level quantum systems (qutrits).
However, using our method we can divide these parameters into four weight, and
four angular parameters, and find that the weight parameters are inducing a
unit sphere in four-dimension. And, the four angular parameters determine
whether a Bloch vector is physical. Therefore, unlike its qubit counterpart,
the qutrit Bloch sphere does not exhibit a solid structure. Importantly, this
construction allows us to define different properties of qutrits in terms of
Bloch vector components. We also examine the two and three-dimensional sections
of the sphere, which reveal a non-convex yet closed structure for physical
qutrit states. Further, we apply our representation to derive mutually unbiased
bases (MUBs), characterize unital maps for qutrits, and assess ensembles using
the Hilbert-Schmidt and Bures metrics. Moreover, we extend this construction to
qudits, showcasing its potential applicability beyond the qutrit scenario.
