On the extreme points of quantum channels

Let L(m,n) denote the convex set of completely positive trace preserving operators from C^{m x m} to C^{n x n}$, i.e quantum channels. We give a necessary condition for L in L(m,n) to be an extreme point. We show that generically, this condition is also sufficient. We characterize completely the extreme points of L_(2,2) and L(3,2), i.e. quantum channels from qubits to qubits and from qutrits to qubits.