This paper considers the problem of efficiently transmitting quantum states through a network. It has been known for some time that without additional assumptions it is impossible to achieve this task perfectly in general -- indeed, it is impossible even for the simple butterfly network. As additional resource we allow free classical communication between any pair of network nodes. It is shown that perfect quantum network coding is achievable in this model whenever classical network coding is possible over the same network when replacing all quantum capacities by classical capacities. More precisely, it is proved that perfect quantum network coding using free classical communication is possible over a network with k source-target pairs if there exists a classical linear (or even vector-linear) coding scheme over a finite ring. Our proof is constructive in that we give explicit quantum coding operations for each network node. This paper also gives an upper bound on the number of classical communication required in terms of k , the maximal fan-in of any network node, and the size of the network.
There are several papers studying quantum network coding on the @math -pair problem in situations different from the most basic setting (perfect transmission of quantum states using only a quantum network of limited capacity). @cite_14 and @cite_9 considered approximate'' transmission of qubits in the @math -pair problem, and showed that transmission with fidelity larger than @math is possible for a class of networks. Hayashi @cite_11 showed how to achieve perfect transmission of two qubits on the butterfly network if two source nodes have prior entanglement, and if, at each edge, we can choose between sending two classical bits and sending one qubit. Leung, Oppenheim and Winter @cite_19 considered various extra resources such as free forward backward two-way classical communication and entanglement, and investigated the lower upper bounds of the rate of quantum network coding for their settings. The setting of the present paper is close to their model allowing free two-way classical communication. The difference is that Ref. @cite_19 considered asymptotically perfect transmission while this paper focuses on perfect transmission. Also, Ref. @cite_19 showed optimal rates for a few classes of networks while the present paper gives lower bounds for much wider classes of networks.
@cite_19 @cite_9 @cite_14 @cite_11
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We study the problem of k-pair communication (or multiple unicast problem) of quantum information in networks of quantum channels. We consider the asymptotic rates of high fidelity quantum communication between specific sender-receiver pairs. Four scenarios of classical communication assistance (none, forward, backward, and two-way) are considered. (I) We obtain outer and inner bounds of the achievable rate regions in the most general directed networks. (II) For two particular networks (including the butterfly network), routing is proved optimal, and the free assisting classical communication can at best be used to modify the directions of quantum channels in the network. Consequently, the achievable rate regions are given by counting edge avoiding paths, and precise achievable rate regions in all four assisting scenarios can be obtained. (III) Optimality of routing can also be proved in classes of networks. The first class consists of directed unassisted networks in which (1) the receivers are information sinks, (2) the maximum distance from senders to receivers is small, and (3) a certain type of 4-cycles are absent, but without further constraints (such as on the number of communicating and intermediate parties). The second class consists of arbitrary backward-assisted networks with two sender-receiver pairs. (IV) Beyond the k-pair communication problem, observations are made on quantum multicasting and a static version of network communication related to the entanglement of assistance. Network coding is often explained by using a small network model called Butterfly. In this network, there are two flow paths, s_1 to t_1 and s_2 to t_2, which share a single bottleneck channel of capacity one. So, if we consider conventional flow (of liquid, for instance), then the total amount of flow must be at most one in total, say 1 2 for each path. However, if we consider information flow, then we can send two bits (one for each path) at the same time by exploiting two side links, which are of no use for the liquid-type flow, and encoding decoding operations at each node. This is known as network coding and has been quite popular since its introduction by Ahlswede, Cai, Li and Yeung in 2000. In QIP 2006, showed that quantum network coding is possible for Butterfly, namely we can send two qubits simultaneously with keeping their fidelity strictly greater than 1 2. In this paper, we show that the result can be extended to a large class of general graphs by using a completely different approach. The underlying technique is a new cloning method called entanglement-free cloning which does not produce any entanglement at all. This seems interesting on its own and to show its possibility is an even more important purpose of this paper. Combining this new cloning with approximation of general quantum states by a small number of fixed ones, we can design a quantum network coding protocol which simulates'' its classical counterpart for the same graph. We find a protocol transmitting two quantum states crossly in the butterfly network only with prior entanglement between two senders. This protocol requires only one qubit transmission or two classical bits (cbits) transmission in each channel in the butterfly network. It is also proved that it is impossible without prior entanglement. More precisely, an upper bound of average fidelity is given in the butterfly network when prior entanglement is not allowed. The presented result concerns only the butterfly network, but our techniques can be applied to a more general graph.