  We describe a general approach to proving the impossibility of implementing a
quantum channel by local operations and classical communication (LOCC), even
with an infinite number of rounds, and find that this can often be demonstrated
by solving a set of linear equations. The method also allows one to design an
LOCC protocol to implement the channel whenever such a protocol exists in any
finite number of rounds. Perhaps surprisingly, the computational expense for
analyzing LOCC channels is not much greater than that for LOCC measurements. We
apply the method to several examples, two of which provide numerical evidence
that the set of quantum channels that are not LOCC is not closed and that there
exist channels that can be implemented by LOCC either in one round or in three
rounds that are on the boundary of the set of all LOCC channels.
  Although every LOCC protocol must implement a separable quantum channel, it
is a very difficult task to determine whether or not a given channel is
separable. Fortunately, prior knowledge that the channel is separable is not
required for application of our method.
