Classic and quantum network coding

Summary form only given. The notion of network coding is often explained by using the butterfly network. In its quantum version, classical channels are replaced by quantum channels with the same capacity. Any unitary operation is allowed at each node. (But note that fan-out is not free in quantum.) From the analogy of the above classical result, the natural attempt is to try sending two quantum bits on the crossing two paths. Unfortunately this seems difficult and it is unlikely to be able to keep a good quality of the transferred quantum bits. However, it turns out that we can keep a reasonably good quality for one qubit in one path and one classical bit in the other and for two special qubits. In the latter case, we can also think of sending four classical bits on the two paths, two for each.