Nowadays, network coding has been a very successful topic in information theory including many applications such as wireless networks while one of the simplest benefits is the efficient transmission in networks by allowing us to encode the transmitted information at every intermediate node. Since quantum information is much more expensive for communication than classical information, it was natural that this benefit made quantum researchers motivate the study of “quantum network coding.” This paper reports the current status of quantum network coding. At present, quantum network coding mostly means sending quantum information (rather than classical information) in a quantum network. Since quantum information cannot be cloned (the quantum no-cloning theorem), multiple unicast networks have been well-studied (in several settings). We present some of the known possibilities and limitations, and future works of quantum network coding, focusing on multiple unicast networks.
Nowadays, network coding has been a very successful topic in information theory including many applications such as wireless networks while one of the simplest benefits is the efficient transmission in networks by allowing us to encode the transmitted information at every intermediate node. Since quantum information is much more expensive for communication than classical information, it was natural that this benefit made quantum researchers motivate the study of “quantum network coding.” This paper reports the current status of quantum network coding. At present, quantum network coding mostly means sending quantum information (rather than classical information) in a quantum network. Since quantum information cannot be cloned (the quantum no-cloning theorem), multiple unicast networks have been well-studied (in several settings). We present some of the known possibilities and limitations, and future works of quantum network coding, focusing on multiple unicast networks.
Considering extensive studies on network coding and quantum information theory in this century, it was just a matter of time that the study of quantum network coding was initiated. In some sense, it is rather important to ask if the amount of quantum bits communicated on a network can be reduced by allowing encoding at intermediate nodes on the network since quantum communication is much more expensive than classical communication in principle. The first result by Hayashi et al. in 2006 was basically negative for this question - quantum network coding does not give any benefit for reducing the amount of qubits sent on the butterfly network. However, this was just the first step to explore the theory of quantum network coding. After their work, it has been shown that the amount of communicated qubits can be reduced in several changes of the settings such as by allowing extra classical communication, while there has been further negative results. In this paper, we review the basic concept of quantum network coding, and report the currently known positive and negative results on quantum network coding in various settings.
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, s1 to t1 and s2 to t2 , 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 ∣ψ1⟩ from s1 to t1 and ∣ψ2⟩ from s2 to t2 simultaneously with a fidelity strictly greater than 1/2.(ii) If one of ∣ψk⟩ and ∣ψ2⟩ is classical, then the fidelity can be improved to 2/3.(iii) Similar improvement is also possible if Cbi⟩ and 1Cb2⟩ 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 s1 to t1 (similarly from s2 to t2 ) but only one of them should be recovered.
Quantum network coding with classical assistance can transmit quantum information with fidelity one in error-free quantum channels. However, the dependency on classical network solutions leads to higher communication complexity and transmission congestion over a large-scale network with bottleneck channels. We find that certain types of quantum operation are sufficient to support multiple unicast sessions over the extended butterfly network. Furthermore, this construction decouples classical communication (assisting decoding information) from classical network solution. Based on this design, we propose an optimal protocol for this specific network, which significantly reduces the communication cost.
This paper considers the communication over a quantum multiple-unicast network where r sender-receiver pairs communicate independent quantum states. We concretely construct a quantum network code for the quantum multiple-unicast network as a generalization of the code [Song and Hayashi, arxiv:1801.03306, 2018] for the quantum unicast network. When the given node operations are restricted to invertible linear operations between bit basis states and the rates of transmissions and interferences are restricted, our code certainly transmits a quantum state for each sender-receiver pair by n-use of the network asymptotically, which guarantees no information leakage to the other users. Our code is implemented only by the coding operation in the senders and receivers and employs no classical communication and no manipulation of the node operations. Several networks that our code can be applied are also given.
Multicasting of quantum states is an essential feature of quantum internet. Since the noncloning theorem prohibits perfect cloning of an unknown quantum state, an appropriate protocol may depend on the purpose of the multicast. In this paper, we treat the multicasting of a single copy of an unknown state over a quantum network with free classical communication. We especially focus on protocols exactly multicasting an asymmetric optimal universal clone. Hence, these protocols are optimal and universal in terms of mean fidelity between input and output states, but the fidelities can depend on target nodes. Among these protocols, a protocol spending smaller communication resources is preferable. Here, we construct such a protocol attaining the min-cut of the network described as follows. Two (three) asymmetric optimal clones of an input state are created at a source node. Then, the state is divided into classical information and a compressed quantum state. The state is sent to two (three) target nodes using the quantum network coding. Finally, the asymmetric clones are reconstructed using LOCC with a small amount of entanglement shared among the target nodes and the classical information sent from the source node.
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.