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. x1 x2 s1 ♠ s2 ♠ ❍❍❍❍ ❥ x1 x2 ✟ n1 x1 x1 ⊕ x2 x2 n2 ♠ x1 ⊕ x2✟ ❍❍❍❍ ❥ x1 ⊕ x2
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. x1 x2 s1 ♠ s2 ♠ ❍❍❍❍ ❥ x1 x2 ✟ n1 x1 x1 ⊕ x2 x2 n2 ♠ x1 ⊕ x2✟ ❍❍❍❍ ❥ x1 ⊕ x2
Considering the problem of how to efficiently transmit unknown quantum states in multi-unicast networks, we propose a nonlinear quantum network coding scheme with classical communication resource. On the conditions of free classical communication between adjacent nodes in a network, as well as unlimited capacities of classical channels, perfect quantum network coding is available. By means of local phase deviation correction, we design the encoding procedure of nonlinear coding scheme. Moreover, we provide the upper bound on the total number of classical communication which is required to achieve quantum communication, as well as the analysis of scheme property.
The k-pair problem in network coding theory asks to send k messages simultaneously between k source-target pairs over a directed acyclic graph. In a previous paper [ICALP 2009, Part I, pages 622-633] the present authors showed that if a classical k-pair problem is solvable by means of a linear coding scheme, then the quantum k-pair problem over the same graph is also solvable, provided that classical communication can be sent for free between any pair of nodes of the graph. Here we address the main case that remained open in our previous work, namely whether nonlinear classical network coding schemes can also give rise to quantum network coding schemes. This question is motivated by the fact that there are networks for which no linear solutions exist to the k-pair problem, whereas nonlinear solutions exist. In the present paper we overcome the limitation to linear protocols and describe a new communication protocol for perfect quantum network coding that improves over the previous one as follows: (i) the new protocol does not put any condition on the underlying classical coding scheme, that is, it can simulate nonlinear communication protocols as well, and (ii) the amount of classical communication sent in the protocol is significantly reduced.
This paper considers a problem of quantum communication between parties that are connected through a network of quantum channels. The model in this paper assumes that there is no prior entanglement shared among any of the parties, but that classical communication is free. The task is to perfectly transfer an unknown quantum state from a source subsystem to a target subsystem, where both source and target are formed by ordered sets of some of the nodes. It is proved that a lower bound of the rate at which this quantum communication task is possible is given by the classical min-cut max-flow theorem of network coding, where the capacities in question are the quantum capacities of the edges of the network.
We study the problem of perfectly multicasting symmetric universal clones of unknown quantum states over quantum networks with free classical communication.
We construct a protocol that multicasts symmetric universal clones of input states from multiple source nodes by extending the quantum network coding protocol proposed by Kobayashi et al. We further establish a sufficient condition for perfect multicast in the single-source setting. Specifically, we show that when a single copy of a qr-dimensional input state is available at the source node, where q is a sufficiently large prime power, perfect multicast of the corresponding symmetric universal clone is achievable using a small amount of entanglement shared among the target nodes.
This result holds for quantum networks represented by an undirected graph G, where each edge corresponds to a noiseless q-dimensional quantum channel, provided that there exists an acyclic directed graph G′ obtained by assigning directions to the edges of G such that the minimum cut of G′ is at least r.
The k-pair problem in network coding theory asks to send k messages simultaneously between k source-target pairs over a directed acyclic graph. In a previous paper [ICALP 2009, Part I, pages 622--633] the present authors showed that if a classical k-pair problem is solvable by means of a linear coding scheme, then the quantum k-pair problem over the same graph is also solvable, provided that classical communication can be sent for free between any pair of nodes of the graph. Here we address the main case that remained open in our previous work, namely whether nonlinear classical network coding schemes can also give rise to quantum network coding schemes. This question is motivated by the fact that there are networks for which there are no linear solutions to the k-pair problem, whereas nonlinear solutions exist. In the present paper we overcome the limitation to linear protocols and describe a new communication protocol for perfect quantum network coding that improves over the previous one as follows: (i) the new protocol does not put any condition on the underlying classical coding scheme, that is, it can simulate nonlinear communication protocols as well, and (ii) the amount of classical communication sent in the protocol is significantly reduced.
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, Hayashi et al 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.