Abstract
Distributing quantum state and entanglement between distant nodes is a crucial task in distributed quantum information processing on largescale quantum networks. Quantum network coding provides an alternative solution for quantumstate distribution, especially when the bottleneck problems must be considered and high communication speed is required. Here, we report the first experimental realization of quantum network coding on the butterfly network. With the help of prior entanglements shared between senders, two quantum states can be transmitted perfectly through the butterfly network. We demonstrate this protocol by employing eight photons generated via spontaneous parametric downconversion. We observe crosstransmission of singlephoton states with an average fidelity of 0.9685 ± 0.0013, and that of twophoton entanglement with an average fidelity of 0.9611 ± 0.0061, both of which are greater than the theoretical upper bounds without prior entanglement.
Introduction
The global quantum network^{1} is believed to be the nextgeneration informationprocessing platform and promises an exponential increase in computation speed, a secure means of communication,^{2,3} and an exponential saving in transmitted information.^{4} The efficient distribution of quantum state and entanglement is a key ingredient for such a global platform. Entanglement distribution^{5} and quantum teleportation^{6} can be employed to transmit quantum states over long distances. By exploiting entanglement swapping^{5} and quantum purification, the transmission distance could be extended significantly, and the fidelities of transmitted states can be enhanced up to unity, which is known as quantum repeaters.^{7} However, with the increased of complexity of quantum networks, especially when many parties require simultaneous communication and communication rates exceed the capacity of quantum channels, low transmission rates, or long delays, known as bottleneck problems, are expected to occur. Thus, it is important to resolve the bottleneck problem and achieve highspeed quantum communication. This question is in the line with issues related to quantum communication complexity, which attempts to reduce the amount of information to be transmitted in solving distributed computational tasks.^{8}
The bottleneck problem is common in classical networks. A landmark solution is the network coding concept,^{9} where the key idea is to allow coding and replication of information locally at any intermediate node in the network. The metadata arriving from two or more sources at intermediate nodes can be combined into a single packet, and this distribution method can increase the effective capacity of a network by minimizing the number and severity of bottlenecks. The improvement is most pronounced when the network traffic volume is near the maximum capacity obtainable via traditional routing. As a result, network coding has realized a new communicationefficient method to send information through networks.^{10}
A primary question relative to quantum networks is whether network coding is possible for quantumstate transmission, which is referred as quantum network coding (QNC). Classical network coding cannot be applied directly in a quantum case due to the nocloning theorem.^{11} However, remarkable theoretical effort has been directed at this important question. For example, Hayashi et al.^{12} were the first to study QNC, and they proved that perfect quantum state crosstransmission is impossible in the butterfly network, i.e., the fidelity of crossly transmitted quantum states cannot reach one. However, if two senders have shared entanglements priorly, the perfect QNC is possible by exploiting quantum teleportation.^{13,14,15} Thus, various studies have focused on network coding for quantum networks, such as the multicast problem,^{16,17} QNC based on quantum repeaters,^{18} QNCbased quantum computation,^{19} and other efficient quantumcommunication protocols with entanglement.^{20,21,22,23} Despite these theoretical advances, to the best of our knowledge, an experimental demonstration of QNC has not been realized in a laboratory, even for the simplest of cases.
In this study, we provide the first experimental demonstration of a perfect QNC protocol on the butterfly network. This experiment adopted the protocol proposed by Hayashi,^{14} who proved that perfect QNC is achievable if the two senders have two prior maximally entangled pairs, while it is impossible without prior entanglement. We demonstrate this protocol by employing eight photons generated via spontaneous parametric downconversion (SPDC). We observed a crosstransmission of singlephoton states with an average fidelity of 0.9685 ± 0.0013, as well as crosstransmission of twophoton entanglement with an average fidelity of 0.9611 ± 0.0061, both of which are greater than the theoretical upper bounds without prior entanglement.
Results
QNC on butterfly network
Network coding refers to coding at a node in a network.^{9} The most famous example of network coding is the butterfly network, which is illustrated in Fig. 1a. While network coding has been generally considered for multicast in a network, its throughput advantages are not limited to multicast. We focus on a simple modification of the butterfly network that facilitates an example involving two simultaneous unicast connections. This is also known as 2pairs problem,^{24,25} which seeks to answer the following: for two sender–receiver pairs (S_{1}–R_{1} and S_{2}–R_{2}), is there a way to send two messages between the two pairs simultaneously? In the network shown in Fig. 1a, each arc represents a directed link that can carry a single packet reliably. Here, is a single packet b_{1} presents at sender S_{1} that we want to transmit to receiver R_{1}, and a single packet b_{2} presents at sender S_{2} that we want to transmit to receiver R_{2} simultaneously. The intermediate node C_{1} breaks from the traditional routing paradigm of packet networks, where intermediate nodes are only permitted to make copies of received packets for output. Intermediate node C_{1} performs a coding operation that takes two received packets, forms a new packet by taking the bitwise sum or XOR), of the two packets, and outputs the resulting packet b_{1} ⊕ b_{2}. Ultimately, R_{1} recovers b_{2} by taking the XOR of b_{1} and b_{1} ⊕ b_{2}, and similarly R_{2} recovers b_{1} by taking the XOR of b_{2} and b_{1} ⊕ b_{2}. Therefore, two unicast connections can be established with coding and cannot without coding.
In the case of quantum 2pairs problem, the model is the same butterfly network (Fig. 1a) with unitcapacity quantum channels and the goal is to send two unknown qubits crossly, i.e., to send ρ_{1} from S_{1} to R_{1} and ρ_{2} from S_{2} to R_{2} simultaneously. However, two rules prevent applying classical network coding directly in the quantum case: (i) an XOR operation for two quantum states is not possible; (ii) an unknown quantum state cannot be cloned exactly. Therefore, it has been proven that the quantum 2pairs problem is impossible.^{12}
Hayashi proposed a protocol that addresses the quantum 2pairs problem by exploition prior entanglements between two senders.^{14} As shown in Fig. 1b, the scheme is a resourceefficient protocol that only requires two preshared pairs of maximally entangled state Φ^{+}〉 between the two senders. Notice that if the sender (S_{1}, S_{2}) nodes and receiver (R_{1}, R_{2}) nodes allow to share prior entanglements, then transmitting classical information with classical network coding can complete the task only by using quantum teleportation.^{13} If free classical between all nodes is not limited, perfect 2pair communication over the butterfly network is possible.^{26} However, we consider a more practical situation that the sender and receiver nodes do not share any prior entanglements. Also, the channel capacity is limited to transmit either one qubit or two classical bits. Hayashi proved that the average fidelity of quantum state transmitted is upper bounded by 0.9504 for singlequbit state, and 0.9256 for entanglement without prior entanglement.^{14} However, with prior entanglement between senders, the average fidelity can reach 1. The protocol is summarized as follows (see Fig. 1b).

1.
S_{1} (S_{2}) applies the Bellstate measurement (BSM) between the transmitted state ρ_{1} (ρ_{2}) and one qubit of Φ^{+}〉. According to the result of BSM m_{1}n_{1} (m_{2}n_{2}), S_{1} (S_{2}) perform the unitary operation \(X^{m_1}Z^{n_1}\) (\(X^{m_2}Z^{n_2}\)) on the other qubit of Φ^{+}〉.

2.
S_{1} (S_{2}) sends the quantum state (after the unitary operation) to R_{2} (R_{1}), and sends the classical bits m_{1}n_{1} (m_{2}n_{2}) to C_{1}. C_{1} performs the XOR on m_{1} and m_{2}, n_{1} and n_{2}, respectively, then sends m_{3} = m_{1} ⊕ m_{2} and n_{3} = n_{1} ⊕ n_{2} to C_{2}. C_{2} makes copies of m_{3}n_{3} and sends them to R_{1} and R_{2}, respectively.

3.
R_{1} and R_{2} recover the quantum states ρ_{1} and ρ_{2} by applying the unitary operation \(X^{m_3}Z^{n_3}\) on their received quantum states.
Experimental realization
We demonstrate the perfect QNC protocol by employing the polarization degree of freedom of photons generated via SPDC. As shown in Fig. 2a, an ultraviolet pulse (with a central wavelength of 390 nm, power of 100 mW, pulse duration of 130 fs, and repetition of 80 MHz) passes through four 2mmlong BBO crystals successively, and generates four maximally entangled photon pairs via SPDC in the form of \(\left {{\mathrm{\Psi }}^ + } \right\rangle _{ij} = 1\sqrt 2 \left( {\left {HV} \right\rangle + \left {VH} \right\rangle } \right)_{ij}\). Here, H (V) denotes the horizontal (vertical) polarization and i, j denote the path modes. Then, we use a Bellstate synthesizer to reduce the frequency correlation between two photons^{27,28} (as shown in Fig. 2b). After the Bellstate synthesizer, Ψ^{+}〉_{ij} is converted to \(\left {{\mathrm{\Phi }}^ + } \right\rangle _{ij} = 1\sqrt 2 \left( {\left {HH} \right\rangle + \left {VV} \right\rangle } \right)_{ij}\). We set narrowband filters with fullwidth at halfmaximum (λ_{FWHM}) of 2.8 and 3.6 nm for the e and oray, respectively, and, with this filter setting, we observe an average twophoton coincidence count rate of 21,000 per second with a visibility of 99.6% in the H(V)〉 basis and visibility of 99.0% in the + (−)〉 basis, from which we calculate the fidelity of prepared entangled photons with an ideal Φ^{+}〉 of 99.3%. We estimate a singlepair generation rate of p ≈ 0.0036, and overall collection efficiency of 28%.
Φ^{+}〉_{12} and Φ^{+}〉_{34} are the two entangled pairs priorly shared between S_{1} and S_{2}, i.e., S_{1} holds photons 1 and 3 and S_{2} holds photons 2 and 4. Φ^{+}〉_{56} and Φ^{+}〉_{78} are held by S_{1} and S_{2}, respectively. Photon 5 is projected on \(\alpha _1^ \ast \left H \right\rangle + \beta _1^ \ast \left V \right\rangle\) to prepare ρ_{1} with ideal form in α_{1}H〉 + β_{1}V〉. Similarly, photon 7 is projected on \(\alpha _2^ \ast \left H \right\rangle + \beta _2^ \ast \left V \right\rangle\) to prepare ρ_{2} with ideal form α_{2}H〉 + β_{2}V〉.
On S_{1}’s side, by finely adjusting the position of the prism on the path of photon 1, we interfere with photons 1 and photon 6 on a polarizing beam splitter (PBS) to realize a Bellstate measurement (BSM). The BSM projects photons 1 and 6 to ψ〉 ∈ {Ψ^{+}〉, Ψ^{−}〉, Φ^{+}〉, Φ^{−}〉}. As the complete BSM is impossible with linear optics, we perform the complete measurements with two setup settings by rotating the angle of the halfwave plate (HWP) on path 6 or 1 before they interfere from 0° to 45°. Note that in each setup, the success probability to identify two of the Bell states is 50%. So, the total success probability is 25% in our experiment. The BSM results (different responses on the four detectors after the interference) are related to two classical bits denoted as m_{1}n_{1} ∈ {00, 01, 10, 11}. According to the BSM results, S_{1} applies the unitary operation U_{1} = \(X^{m_1}Z^{n_1}\) on photon 3, and then sends m_{1}n_{1} to node C_{1} and photon 3 to the receiver node R_{2}. Here, we use X, Y, Z to represent the PauliX, PauliY, and PauliZ matrix. Similarly, on the S_{2} side, we interfere with photons 4 and 8 on a PBS to realize a BSM with result of m_{2}n_{2}, according to which S_{2} applies the unitary operation U_{2} = \(X^{m_2}Z^{n_2}\) on photon 2. Then, S_{2} sends m_{2}n_{2} to node C_{1} and sends photon 2 to the receiver node R_{1}.
On node C_{1}, we perform the XOR operation on m_{1} and m_{2} and n_{1} and n_{2}, and send the results m_{3} = m_{1} ⊕ m_{2}, n_{3} = n_{1} ⊕ n_{2} to node C_{2}, where we make two copies of m_{3}n_{3} and send these copies to R_{1} and R_{2}. Finally, according to m_{3}n_{3}, we apply the unitary operation U_{3} = \(X^{m_3}Z^{n_3}\) on photons 3 and photon 2 to recover ρ_{2} and ρ_{1}.
In our experiment, the unitary operation is realized by HWPs with transformation matrix \(U(\theta ) = \left( {\begin{array}{*{20}{c}} {{\mathrm{cos2}}\theta } & {{\mathrm{sin2}}\theta } \\ {{\mathrm{sin2}}\theta } & {  {\mathrm{cos2}}\theta } \end{array}} \right)\), where θ is the angle fast axis relative to the vertical axis. X^{0}Z^{0} = I means no operation on the photon. Here, X^{0}Z^{1} = Z is realized by setting an HWP at 0°. X^{1}Z^{0} = X is realized by setting an HWP at 45°, and X^{1}Z^{1} = XZ is realized by setting two HWPs (one at 45° and the other at 0° (shown in Fig. 2c)).
Experimental results
We first show that two singlephoton states can be crossly delivered from S_{1} to R_{2} and from S_{2} to R_{1} simultaneously in the butterfly network. S_{1} and S_{2} can prepare six individual quantum states ρ_{1} and ρ_{2} with an average fidelity of 99.3%. ρ_{1} and ρ_{2} have an ideal form of ρ_{1} = ϕ_{1}〉〈ϕ_{1} and ρ_{2} = ϕ_{2}〉〈ϕ_{2}, where \(\left {\phi _1} \right\rangle ,\left {\phi _2} \right\rangle\) ∈ \(\left\{ {\left H \right\rangle ,\left V \right\rangle ,\left \pm \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left H \right\rangle \pm \left V \right\rangle } \right),\left {L(R)} \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left H \right\rangle \pm i\left V \right\rangle } \right)} \right\}\). In our experiment, both S_{1} and S_{2} irrelatively select ρ_{1} and ρ_{2} from six states for transmission, thereby resulting in a total of 36 combinations. After recover of R_{1} and R_{2}, we measure the fidelities between the recovered state \(\rho _1^\prime\) (\(\rho _2^\prime\)) and the ideal input state ρ_{1} = ϕ_{1}〉〈ϕ_{1} (ρ_{2} = ϕ_{2}〉〈ϕ_{2}), i.e., \(F_{S_1 \to R_2} = Tr\left( {\left {\phi _1} \right\rangle \left\langle {\phi _1} \right\rho _1^\prime } \right)\) and \(F_{S_2 \to R_1} = Tr\left( {\left {\phi _2} \right\rangle \left\langle {\phi _2} \right\rho _2^\prime } \right)\). We project the photon on the ϕ〉(ϕ_{⊥}〉) basis and record the counts N_{+} and N_{−}, where ϕ_{⊥}〉 is the orthogonal state of ϕ〉. Thus, the fidelity of the transferred singlephoton state can be calculated by \(F = \frac{{N_ + }}{{N_ + + N_  }}\). The average fidelities over all possible BSM outcomes are shown in Fig. 3a. Note that each BSM has four possible outcomes, thus there are 16 combinations of outcomes for the two BSMs. For each combination, we apply the unitary operations and record the measured fidelities. In Fig. 3a, the red line represents the theoretical upper bound of the average fidelity without prior entanglement, i.e., F_{th} = 0.9503. Specifically, Fig. 3b shows the histogram of all measured fidelities of the 576 situations, and the average fidelity is quantified as \(\bar F = \mathop {\sum}\nolimits_i {p_i} F_i = 0.9685 \pm 0.0013\), where p_{i} and F_{i} are the probability and fidelity shown in Fig. 3b. The average fidelity beyonds F_{th} = 0.9503 with 14 standard deviations.
We also show that twophoton entanglement can be established crossly with this setup, i.e., twophoton entanglement can be established between S_{1} and R_{2} and S_{2} and R_{1}, simultaneously. Here, the experimental setup is the same, S_{1}(S_{2}) does not project photon 5(7) on α^{*}H〉 + β^{*}V〉, and photon 5(7) is retained to perform the joint measurements with photon 2(3). To quantify the crossentanglement between photons 5 and 2 and 7 and 3, we measure the entanglement witness on rho_{52} and ρ_{73}, respectively. In particular, we measure the entanglement witness 〈W〉 = I/2 − Φ^{+}〉〈Φ^{+}, which can also be related to the entanglement fidelity 〈W〉 = 1/2 − F_{ent}. Here, F_{ent} is defined as the entanglement fidelity between the entanglement state ρ_{ij} and the maximal entanglement state Φ^{+}〉, i.e., F_{ent} = Tr(ρ_{ij}Φ^{+}〉〈Φ^{+}). Φ^{+}〉〈Φ^{+} can be decomposed to local observables as \(\left {{\mathrm{\Phi }}^ + } \right\rangle \left\langle {{\mathrm{\Phi }}^ + } \right = \frac{{II + XX  YY + ZZ}}{4}\). By measuring the expected values of local observables, we can calculate the entanglement fidelity. The local observable \({\cal{O}}\) can be expressed as \({\cal{O}} = \left \phi \right\rangle \left\langle \phi \right  \left {\phi _ \bot } \right\rangle \left\langle {\phi _ \bot } \right\), where ϕ〉(ϕ_{⊥}〉) is the eigenstate of \({\cal{O}}\) with eigenvalue of 1(−1). The expected value of \({\cal{O}}\) can be calculated by the counts \(\langle {\cal{O}}\rangle = \frac{{N_ +  N_  }}{{N_ + + N_  }}\). The experimental results of the fidelities of crossentanglement are shown in Fig. 4. We calculate that the average fidelity of two crossly established entanglement is 0.9611 ± 0.0061, which beyonds 0.9256 with 5.8 standard deviations.
Discussion
QNC provides an alternative solution for the transition of quantum states in quantum networks. Compared to entanglement swapping, QNC demonstrates superiority especially when quantum resources are limited or a high communication rate is required.^{29} In addition, largescale QNC demonstrates superiority relative to fidelity performance as well.^{30} In this paper, We have demonstrated the first perfect QNC on a butterfly network. The average fidelities of crossstate transmission and crossentanglement distribution achieved in our experiment exceed the theoretical upper bounds permitted without prior entanglement. We expect that our results will pave the way to experimentally explore the advanced features of prior entanglement in quantum communication. In addition, we expect that our results will realize opportunities for various studies of efficient quantum communication protocols in quantum networks with complex topologies.
Data availability
The data are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the National Key Research and Development (R&D) Plan of China (grants 2018YFB0504300 and 2018YFA0306501), the National Natural Science Foundation of China (grants 11425417, 61771443 and 11975222), the Anhui Initiative in Quantum Information Technologies and the Chinese Academy of Sciences. H. Lu was partially supported by Major Program of Shandong Province Natural Science Foundation (grants ZR2018ZB0649). F. Xu thanks Prof. Bin Li for the inspiration to the subject.
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H.L., F.X., Y.A.C., and J.W.P. established the theory and designed the experimental setup. H.L., Z.D.L., Y.X.F., and R.Z. performed the experiment. H.L. X.X.F., L.L., and N.L.L. analyzed the data. H.L., F.X., and Y.A.C. wrote the paper with contributions from all authors. F.X., Y.A.C., and J.W.P. supervised the project.
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Lu, H., Li, ZD., Yin, XF. et al. Experimental quantum network coding. npj Quantum Inf 5, 89 (2019). https://doi.org/10.1038/s4153401902072
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