On the extreme points of quantum channels

Let Lm,n denote the convex set of completely positive trace preserving operators from Cm×m to Cn×n, i.e. quantum channels. We give a necessary condition for L∈Lm,n to be an extreme point. We show that generically, this condition is also sufficient. We characterize completely the extreme points of L2,2 and L3,2, i.e. quantum channels from qubits to qubits and from qutrits to qubits.