Probabilistic quantum network coding of MM-qudit states over the butterfly network

Following some ideas of the quantum network coding protocol [Phys. Rev. A 76 (2007) 040301], we propose an efficient protocol to transmit two 2-level quantum states crossly over the butterfly network with two non-maximally entangled qubit pairs shared only between two senders. It is shown that both of the receivers can reestablish the initial states with a certain probability. And one classical bit communication can be saved at the bottleneck channel. Moreover, this protocol is generalized to the case of transmitting two multipartite entangled states.