  In the Bloch sphere based representation of qudits with dimensions greater
than two, the Heisenberg-Weyl operator basis is not preferred because of
presence of complex Bloch vector components. We try to address this issue and
parametrize a qutrit using the Heisenberg-Weyl operators by identifying eight
real parameters and separate them as four weight and four angular parameters
each. The four weight parameters correspond to the weights in front of the four
mutually unbiased bases sets formed by the eigenbases of Heisenberg-Weyl
observables and they form a four-dimensional unit radius Bloch hypersphere.
Inside the four-dimensional hypersphere all points do not correspond to a
physical qutrit state but still it has several other features which indicate
that it is a natural extension of the qubit Bloch sphere. We study the purity,
rank of three level systems, orthogonality and mutual unbiasedness conditions
and the distance between two qutrit states inside the hypersphere. We also
analyze the two and three-dimensional sections centered at the origin which
gives a close structure for physical qutrit states inside the hypersphere.
Significantly, we have applied our representation to find mutually unbiased
bases(MUBs) and to characterize the unital maps in three dimensions. It should
also be possible to extend this idea in higher dimensions.
