Quantum Network Coding

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> from s_1 to t_1 and |psi_2> from s_2 to t_2 simultaneously with a fidelity strictly greater than 1/2. (ii) If one of |psi_1> and |psi_2> is classical, then the fidelity can be improved to 2/3. (iii) Similar improvement is also possible if |psi_1> and |psi_2> 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.Comment: 27pages, 11figures. The 12page version will appear in 24th International Symposium on Theoretical Aspects of Computer Science (STACS 2007

(4, 1)-Quantum Random Access Coding Does Not Exist

“…As shown in the figure, we can transmit two bits, x and y, on the two paths simultaneously. The paper [9] extends this network coding for quantum channels and quantum information. Their results include; (i) One can send any quantum state |ψ 1 from s 1 to t 1 and |ψ 2 from s 2 to t 2 simultaneously with a fidelity strictly greater than 1/2.…”

Network Coding Applications

“…Quantum multicast was studied by Shi and Soljanin [77], who showed that lossless compression of special multicast quantum states is possible and significantly reduces the edge capacity requirements of the multicast. Two unicasts on the butterfly network of quantum channels were first considered by Hayashi et al [32], who showed that two qubits can be sent simultaneously with keeping their fidelity strictly greater than 1/2 if the nodes of the network can use one of the available algorithms for approximate cloning [7]. This approach was later extended to probabilistic cloning and general graphs by Iwama et al [38].…”

Analysis of quantum network coding for realistic repeater networks

“…It is expected that network coding also allows the same resolution in a quantum network. In recent years, a number of researchers have studied quantum network coding [21][22][23][24][25][26][27] . However, all of these studies presuppose the use of pure states and perfect local gates.…”