  Quantum channels, which are completely positive and trace preserving
mappings, can alter the dimension of a system; e.g., a quantum channel from a
qubit to a qutrit. We study the convex set properties of dimension-altering
quantum channels, and particularly the channel decomposition problem in terms
of convex sum of extreme channels. We provide various quantum circuit
representations of extreme and generalized extreme channels, which can be
employed in an optimization to approximately decompose an arbitrary channel.
Numerical simulations of low-dimensional channels are performed to demonstrate
our channel decomposition scheme.
