  By means of a well-grounded mapping scheme linking Schwinger unitary
operators and generators of the special unitary group $\mathrm{SU(N)}$, it is
possible to establish a self-consistent theoretical framework for
finite-dimensional discrete phase spaces which has the discrete
$\mathrm{SU(N)}$ Wigner function as a legitimate by-product. In this paper, we
apply these results with the aim of putting forth a detailed study on the
discrete $\mathrm{SU(2)} \otimes \mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner
functions, in straight connection with experiments involving, among other
things, the tomographic reconstruction of density matrices related to the
two-qubit and ququart states. Next, we establish a formal correspondence
between both the descriptions that allows us to visualize the quantum
correlation effects of these states in finite-dimensional discrete phase
spaces. Moreover, we perform a theoretical investigation on the two-qubit
X-states, which combines discrete Wigner functions and their respective
marginal distributions in order to obtain a new function responsible for
describing qualitatively the quantum correlation effects. To conclude, we also
discuss possible extensions to the discrete Husimi and Glauber-Sudarshan
distribution functions, as well as future applications on spin chains.
