  Quantum channels can be described via a unitary coupling of system and
environment, followed by a trace over the environment state space. Taking the
trace instead over the system state space produces a different mapping which we
call the conjugate channel. We explore the properties of conjugate channels and
describe several different methods of construction. In general, conjugate
channels map M_d to M_d' with d < d', and different constructions may differ by
conjugation with a partial isometry.
  We show that a channel and its conjugate have the same minimal output entropy
and maximal output p-norm. It then follows that the additivity and
multiplicativity conjectures for these measures of optimal output purity hold
for a product of channels if and only if they also hold for the product of
their conjugates. This allows us to reduce these conjectures to the special
case of maps taking M_d to M_d' with a minimal representation of dimension at
most d.
  We find explicit expressions for the conjugates for a number of well-known
examples, including entanglement-breaking channels, unital qubit channels, the
depolarizing channel, and a subclass of random unitary channels. For the
entanglement-breaking channels, channels this yields a new class of channels
for which additivity and multiplicativity of optimal output purity can be
established. For random unitary channels using the generalized Pauli matrices,
we obtain a new formulation of the multiplicativity conjecture. The conjugate
of the completely noisy channel plays a special role and suggests a mechanism
for using noise to transmit information.
