Probabilistic quantum network coding of M-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. (C) 2009 Elsevier B.V. All rights reserved.