  In quantum Shannon theory, the way information is encoded and decoded takes
advantage of the laws of quantum mechanics, while the way communication
channels are interlinked is assumed to be classical. In this Letter we relax
the assumption that quantum channels are combined classically, showing that a
quantum communication network where quantum channels are combined in a
superposition of different orders can achieve tasks that are impossible in
conventional quantum Shannon theory. In particular, we show that two identical
copies of a completely depolarizing channel become able to transmit information
when they are combined in a quantum superposition of two alternative orders.
This finding runs counter to the intuition that if two communication channels
are identical, using them in different orders should not make any difference.
The failure of such intuition stems from the fact that a single noisy channel
can be a random mixture of elementary, non-commuting processes, whose order (or
lack thereof) can affect the ability to transmit information.
