Quantum Network Coding

Since quantum information is continuous, its handling is sometimes surprisingly harder than the classical counterpart. A typical example is cloning; making a copy of digital information is straightforward but it is not possible exactly for quantum information. The question in this paper is whether or not {em quantum} network coding is possible. Its classical counterpart is another good example to show that digital information flow can be done much more efficiently than conventional (say, liquid) flow. Our answer to the question is similar to the case of cloning, namely, it is shown that quantum network coding is possible if approximation is allowed, by using a simple network model called Butterfly. In this network, there are two flow paths, $s_1$ to $t_1$ and $s_2$ to $t_2$, which shares a single bottleneck channel of capacity one. In the classical case, we can send two bits simultaneously, one for each path, in spite of the bottleneck. Our results for quantum network coding include: (i) We can send any quantum state $|psi_1 angle$ from $s_1$ to $t_1$ and $|psi_2 angle$ from $s_2$ to $t_2$ simultaneously with a fidelity strictly greater than $1/2$. (ii) If one of $|psi_1 angle$ and $|psi_2 angle$ is classical, then the fidelity can be improved to $2/3$. (iii) Similar improvement is also possible if $|psi_1 angle$ and $|psi_2 angle$ are restricted to only a finite number of (previously known) states. (iv) Several impossibility results including the general upper bound of the fidelity are also given.