The notion of network coding is often explained by using the butterfly network (Fig. 1). Every channel has a capacity of one. [Ahlswede, Cai, Li, and Yeung 00] shows that two classical bits can be sent from s1 to t1 and s2 to t2, one for each. It seems that two bits can go through the channel (s0, t0) of capacity one, which is counter-intuitive against the case of "liquid flow". The trick is that any Boolean operation (including free fan-out) is allowed at each node.

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 transfered 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.