Experimental quantum network coding

Distributing quantum state and entanglement between distant nodes is a crucial task in distributed quantum information processing on large-scale quantum networks. Quantum network coding provides an alternative solution for quantum-state 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 cross-transmission of single-photon states with an average fidelity of 0.9685 ± 0.0013, and that of two-photon 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 next-generation information processing platform and promises an exponential increase in computation speed, a secure means of communication 2,3 and an exponential saving in transmitted information. 4he 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. 7However, 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 high-speed 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 to solve distributed computational tasks. 8he 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 communication-efficient method to send information through networks. 10 primary question relative to quantum networks is whether network coding is possible for quantum state transmission, which is referred as quantum network coding (QNC).Classical network coding cannot be applied directly in a quantum case due to the no-cloning theorem. 11However, 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 cross-transmission is impossible in the butterfly network, i.e., the fidelity of crossly transmitted quantum states cannot reach one.1][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 down-conversion (SPDC).We observed a cross-transmission of single-photon states with an average fidelity of 0.9685 ± 0.0013, as well as cross-transmission of two-photon entanglement with an average fidelity of 0.9611 ± 0.0061, both of which are greater than the theoretical upper bounds without prior entanglement.

QNC on butterfly network
Network coding refers to coding at a node in a network. 9The 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 2-pairs 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 present at sender S 1 that we want to transmit to receiver R 1 and a single packet b 2 present at sender S 2 that we want to transmit to receiver R In the case of quantum 2-pairs problem, the model is the same butterfly network (Fig. 1a) with unit-capacity 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 2-pairs problem is impossible. 12ayashi proposed a protocol that addresses the quantum 2-pairs problem by exploition prior entanglements between two senders. 14As shown in Fig. 1b, the scheme is a resource-efficient protocol that only requires two pre-shared 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   2b). to improve the counter rate of entangled photon pair.To avoiding a mess of illustration, we separated propagation of ultraviolet pulse.In our experiment, the ultraviolet pulse is guided by mirrors to shine on four BBO one by one.All the photons are collected by single-mode fiber and detected by single photon detecters (SPD).The coincidence is recored by several home-made field-programmable gate arrays (FPGA).See main text for more details.b, Bell-state synthesizer.The generated photons are compensated by a HWP at 45 • and 1-mm-long BBO crystal.Then, one photon is rotated by a HWP at 45 • and finally two photons are recombined on a PBS.With Bell-state synthesizer makes ordinary ray(o-ray) exiting from one port of PBS and extraordinary ray(e-ray) exiting the other port of PBS.c, the unitary operation U i = X m i Z n i is realized by HWPs.We post-selectively apply U i according to m i n i .c, symbols used in a, b and c.BBO: Beta barium borate crystal.PBS: Polarizing beam splitter.HWP: Half-wave plate.QWP: Quarter-wave plate.SPD: Single photon detector.
share prior entanglements, then transmitting classical information with classical network coding can complete the task only by using quantum teleportation. 13If free classical between all nodes is not limited, perfect 2-pair communication over the butterfly network is possible. 26However, 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 single-qubit state and 0.9256 for entanglement without prior entanglement. 14However, 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 Bell-state 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 + .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) successively passes through four 2-mm-long BBO crystals successively and generates four maximally entangled photon pairs via SPDC in the form of Here H (V ) denotes the horizontal (vertical) polarization and i, j denote the path modes.Then, we use a Bell-state synthesizer to reduce the frequency correlation between two photons 27,28 (as shown in Fig. 2b).After the Bell-state synthesizer, |Ψ + i j is converted to We set narrow-band filters with full-width at half maximum (λ FW HM ) of 2.8 nm and 3.6 nm for the e-and o-ray, respectively, and, with this filter setting, we observe an average two-photon coincidence count rate of 21000 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 single-pair generation rate of p ≈ 0.0036 and overall collection efficiency of 28%.
|Φ |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 Bell-state measurement (BSM).The BSM projects photons 1 and photon 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 half-wave 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 Pauli-X, Pauli-Y , Pauli-Z 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 .

Figure 1 .
Figure 1.Classical network coding and quantum network coding on a butterfly network.a, classical network coding on a butterfly network.Dash line with arrow represents information flow with a capacity of a single packet.In the two simultaneous unicast connections problem, one packet b 1 presented at source node S 1 is required to transmit to node R 1 and the other packet b 2 presented at source node S 2 is required to transmit to node R 2 simultaneously.The intermediate node C 1 performs a coding operation XOR ⊕ on b 1 and b 2 .C 2 makes copies of b 1 ⊕ b 2 and sends them to R 1 and R 2 respectively.R 1 and R 2 decode by performing further XOR operations on the packets that they each receive.b, quantum network coding on butterfly network.The red line with arrow represents quantum information flow with a capacity of a single qubit, and the dash line with arrow represents classical information flow with a capacity of a two bits.See main text for more details.
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.

Figure 2 .
Figure 2. Schematic drawing of the experimental setup.a, an ultraviolet pulse successively pass through four BBO crystals, and generate four pairs of maximally entangled photons.We use four Bell-state synthesizer (shown in Fig.2b).to improve the counter rate of entangled photon pair.To avoiding a mess of illustration, we separated propagation of ultraviolet pulse.In our experiment, the ultraviolet pulse is guided by mirrors to shine on four BBO one by one.All the photons are collected by single-mode fiber and detected by single photon detecters (SPD).The coincidence is recored by several home-made field-programmable gate arrays (FPGA).See main text for more details.b, Bell-state synthesizer.The generated photons are compensated by a HWP at 45 • and 1-mm-long BBO crystal.Then, one photon is rotated by a HWP at 45 • and finally two photons are recombined on a PBS.With Bell-state synthesizer makes ordinary ray(o-ray) exiting from one port of PBS and extraordinary ray(e-ray) exiting the other port of PBS.c, the unitary operation U i = X m i Z n i is realized by HWPs.We post-selectively apply U i according to m i n i .c, symbols used in a, b and c.BBO: Beta barium borate crystal.PBS: Polarizing beam splitter.HWP: Half-wave plate.QWP: Quarter-wave plate.SPD: Single photon detector.

2 ./ 7 HHFigure 3 .
Figure 3. Fidelities of crossly transmitted quantum states.a, The green bar represents F S 1 →R 2 , and the yellow bars represents F S 2 →R 1 .The pair-appeared bars represent fidelities measured simultaneously at R 1 and R 2 .For example, HR means that S 1 delivers |H and S 2 delivers |R .The red line represents the threshold of F t h = 0.9503.The fourfold coincidence is approximately 1.5 counts per second.We accumulate coincidences for 240 seconds and a total of 720 counts for each two-state transition.The error bars are calculated assuming a Poisson statistics for the coincidence counts and Gaussian error propagation.b, Histogram of state fidelities.