  As the quantum analog of the classical one-time pad, the private quantum
channel (PQC) plays a fundamental role in the construction of the maximally
mixed state (from any input quantum state), which is very useful for studying
secure quantum communications and quantum channel capacity problems. However,
the undoubted existence of a relation between the geometric shape of regular
polytopes and private quantum channels in the higher dimension has not yet been
reported. Recently, it was shown that a one-to-one correspondence exists
between single-qubit PQCs and three-dimensional regular polytopes (i.e.,
regular polyhedra). In this paper, we highlight these connections by exploiting
two strategies known as a generalized Gell-Mann matrix and modified quantum
Fourier transform. More precisely, we explore the explicit relationship between
PQCs over a qutrit system (i.e., a three-level quantum state) and regular
4-polytope. Finally, we attempt to devise a formula for connections on higher
dimensional cases.
