Efficient Transmission of Information on the Quantum Network(New Trends in Theory of Computation and Algorithm)

Abstract

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 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 $|\psi_{1}\rangle$ from $s_{1}$ to $t_{1}$ and $|\psi_{2}\rangle$ from s2 to $t_{2}$ simultaneously with a fidelity strictly greater than 1/2.(ii) If one of $|\psi_{\mathrm{k}}\rangle$ and $|\psi_{2}\rangle$ is classical, then the fidelity can be improved to 2/3.(iii) Similar improvement is also possible if $\mathrm{C}\mathrm{b}_{\mathrm{i}}\rangle$ and $1\mathrm{C}\mathrm{b}2\rangle$ are restricted to only a finite number of (previously known) states.This allows us to design an interesting protocol which can send two classical bits from $s_{1}$ to $t_{1}$ (similarly from $s_{2}$ to $t_{2}$ ) but only one of them should be recovered.