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arXiv:1205.3745 · Physical Review A

Quantum network coding for quantum repeaters

Takahiko Satoh, François Le Gall, Hiroshi Imai

Published Sep 24, 2012• 78 citations
arXivHTMLPDF

Abstract

This paper considers quantum network coding, which is a recent technique that enables quantum information to be sent on complex networks at higher rates than straightforward routing strategies. Kobayashi et al. have recently showed the potential of this technique by demonstrating how any classical network coding protocol gives rise to a quantum network coding protocol. They nevertheless primarily focused on an abstract model, in which quantum resources such as additional quantum registers can be freely introduced at each node. In this work, we present a protocol for quantum network coding under weaker (and more practical) assumptions: Our new protocol works even for quantum networks where adjacent nodes initially share one Einstein-Podolsky-Rosen pair but cannot add any additional quantum registers or send any quantum information. A typical example of networks satisfying this assumption is quantum repeater networks, which are promising candidates for the implementation of large-scale quantum networks. Our results thus show that quantum network coding techniques can increase the transmission rate in such quantum networks as well.

DOI
10.1103/physreva.86.032331

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This paper considers quantum network coding, which is a recent technique that enables quantum information to be sent on complex networks at higher rates than straightforward routing strategies. Kobayashi et al. have recently showed the potential of this technique by demonstrating how any classical network coding protocol gives rise to a quantum network coding protocol. They nevertheless primarily focused on an abstract model, in which quantum resources such as additional quantum registers can be freely introduced at each node. In this work, we present a protocol for quantum network coding under weaker (and more practical) assumptions: Our new protocol works even for quantum networks where adjacent nodes initially share one Einstein-Podolsky-Rosen pair but cannot add any additional quantum registers or send any quantum information. A typical example of networks satisfying this assumption is quantum repeater networks, which are promising candidates for the implementation of large-scale quantum networks. Our results thus show that quantum network coding techniques can increase the transmission rate in such quantum networks as well.
Quantum repeater networks have attracted attention for the implementation of long-distance and large-scale sharing of quantum states. Recently, researchers extended classical network coding, which is a technique for throughput enhancement, into quantum information. The utility of quantum network coding (QNC) has been shown under ideal conditions, but it has not been studied previously under conditions of noise and shortage of quantum resources. We analyzed QNC on a butterfly network, which can create end-to-end Bell pairs at twice the rate of the standard quantum network repeater approach. The joint fidelity of creating two Bell pairs has a small penalty for QNC relative to entanglement swapping. It will thus be useful when we care more about throughput than fidelity. We found that the output fidelity drops below 0.5 when the initial Bell pairs have fidelity F<0.90F<0.90F<0.90, even with perfect local gates. Local gate errors have a larger impact on quantum network coding than on entanglement swapping.
Abstract Remote quantum entanglement can enable numerous applications including distributed quantum computation, secure communication, and precision sensing. We consider how a quantum network—nodes equipped with limited quantum processing capabilities connected via lossy optical links—can distribute high-rate entanglement simultaneously between multiple pairs of users. We develop protocols for such quantum “repeater” nodes, which enable a pair of users to achieve large gains in entanglement rates over using a linear chain of quantum repeaters, by exploiting the diversity of multiple paths in the network. Additionally, we develop repeater protocols that enable multiple user pairs to generate entanglement simultaneously at rates that can far exceed what is possible with repeaters time sharing among assisting individual entanglement flows. Our results suggest that the early-stage development of quantum memories with short coherence times and implementations of probabilistic Bell-state measurements can have a much more profound impact on quantum networks than may be apparent from analyzing linear repeater chains. This framework should spur the development of a general quantum network theory, bringing together quantum memory physics, quantum information theory, quantum error correction, and computer network theory.
Quantum network coding can effectively improve the aggregate throughput of quantum networks and alleviate bottlenecks caused by topological constraints. Most of previous schemes are dedicated to the efficient teleportation of unknown quantum states in a quantum network. Herein a proposal for transmission of deterministic known states over quantum repeater network based on quantum measurements. We show that the new protocol offers advantages over three aspects. Firstly, the senders in our protocol obtain the knowledge of the quantum state to be transmitted, which enables the autonomy of quantum network transmission. Secondly, we study the quantum repeater network coding for long-distance deterministic quantum state communication. Quantum repeater network initialization requires entanglement distribution only among neighboring nodes, greatly saving entanglement resources, channel overhead and storage costs. Thirdly, based on Pauli measurements and local complementation, new protocol realizes parallel coding operations to mitigate latency issues sufficiently. Combining quantum network coding and quantum remote state preparation technology, our protocol provides an important solution for deterministic known states transmission over large-scale quantum network in the future.
Network coding is a technique to maximize communication rates within a network, in communication protocols for simultaneous multi-party transmission of information. Linear network codes are examples of such protocols in which the local computations performed at the nodes in the network are limited to linear transformations of their input data (represented as elements of a ring, such as the integers modulo 2). The quantum linear network coding protocols of Kobayashi et al. coherently simulate classical linear network codes, using supplemental classical communication. We demonstrate that these protocols correspond in a natural way to measurement-based quantum computations with graph states over qudits having a structure directly related to the network.
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, s1s_{1}s1​ to t1t_{1}t1​ and s2s_{2}s2​ to t2t_{2}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⟩|\psi_{1}\rangle∣ψ1​⟩ from s1s_{1}s1​ to t1t_{1}t1​ and ∣ψ2⟩|\psi_{2}\rangle∣ψ2​⟩ from s2 to t2t_{2}t2​ simultaneously with a fidelity strictly greater than 1/2.(ii) If one of ∣ψk⟩|\psi_{\mathrm{k}}\rangle∣ψk​⟩ and ∣ψ2⟩|\psi_{2}\rangle∣ψ2​⟩ is classical, then the fidelity can be improved to 2/3.(iii) Similar improvement is also possible if Cbi⟩\mathrm{C}\mathrm{b}_{\mathrm{i}}\rangleCbi​⟩ and 1Cb2⟩1\mathrm{C}\mathrm{b}2\rangle1Cb2⟩ 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 s1s_{1}s1​ to t1t_{1}t1​ (similarly from s2s_{2}s2​ to t2t_{2}t2​ ) but only one of them should be recovered.
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.