  The operator-sum decomposition (OS) of a mapping from one density matrix to
another has many applications in quantum information science. To this mapping
there corresponds an affine map which provides a geometric description of the
density matrix in terms of the polarization vector representation. This has
been thoroughly explored for qubits since the components of the polarization
vector are measurable quantities (corresponding to expectation values of
Hermitian operators) and also because it enables the description of map domains
geometrically. Here we extend the OS-affine map correspondence to qudits,
briefly discuss general properties of the map, the form for particular
important cases, and provide several explicit results for qutrit maps. We use
the affine map and a singular-value-like decomposition, to find positivity
constraints that provide a symmetry for small polarization vector magnitudes
(states which are closer to the maximally mixed state) which is broken as the
polarization vector increases in magnitude (a state becomes more pure). The
dependence of this symmetry on the magnitude of the polarization vector implies
the polar decomposition of the map can not be used as it can for the qubit
case. However, it still leads us to a connection between positivity and purity
for general d-state systems.
