Quantum network coding and the current status of its studies

Considering extensive studies on network coding and quantum information theory in this century, it was just a matter of time that the study of quantum network coding was initiated. In some sense, it is rather important to ask if the amount of quantum bits communicated on a network can be reduced by allowing encoding at intermediate nodes on the network since quantum communication is much more expensive than classical communication in principle. The first result by Hayashi et al. in 2006 was basically negative for this question - quantum network coding does not give any benefit for reducing the amount of qubits sent on the butterfly network. However, this was just the first step to explore the theory of quantum network coding. After their work, it has been shown that the amount of communicated qubits can be reduced in several changes of the settings such as by allowing extra classical communication, while there has been further negative results. In this paper, we review the basic concept of quantum network coding, and report the currently known positive and negative results on quantum network coding in various settings.