  The set of all channels with fixed input and output is convex. We first give
a convenient formulation of necessary and sufficient condition for a channel to
be extreme point of this set in terms of complementary channel, a notion of big
importance in quantum information theory. This formulation is based on the
general approach to extremality of completely positive maps in an operator
algebra due to Arveson. We then apply this formulation to prove the main result
of this note: under certain nondegeneracy conditions, purity of the environment
is necessary and sufficient for extremality of Bosonic linear (quasi-free)
channel. It follows that Gaussian channel between finite-mode Bosonic systems
is extreme if and only if it has minimal noise.
