Experimental Quantum Network Coding
He Lu, Zheng-Da Li, Xu-Fei Yin, Rui Zhang, Xiao-Xu Fang, Li Li, Nai-Le Liu, Feihu Xu, Yu-Ao Chen, Jian-Wei Pan
EExperimental Quantum Network Coding
He Lu , Zheng-Da Li , Xu-Fei Yin , Rui Zhang , Xiao-Xu Fang , Li Li , Nai-LeLiu , Feihu Xu , Yu-Ao Chen , and Jian-Wei Pan Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Shanghai 201315, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science andTechnology of China, Hefei, Anhui 230026, China School of Physics, Shandong University, Jinan 250100, China * Correspondence: Feihu Xu ([email protected]); Yu-Ao Chen ([email protected]); Jian-Wei Pan([email protected])
ABSTRACT
Distributing quantum state and entanglement between distant nodes is a crucial task in distributed quantum informationprocessing on large-scale quantum networks. Quantum network coding provides an alternative solution for quantum statedistribution especially when the bottleneck problems must be considered and high communication speed is required. Here, wereport the first experimental realization of quantum network coding on the butterfly network. With the help of prior entanglementsshared between senders, two quantum states can be transmitted perfectly through the butterfly network. We demonstrate thisprotocol by employing eight photons generated via spontaneous parametric down-conversion. We observe cross-transmissionof single-photon states with an average fidelity of . ± . , and that of two-photon entanglement with an average fidelityof . ± . , both of which are greater than the theoretical upper bounds without prior entanglement. Introduction
The global quantum network is believed to be the next-generation information processing platform and promises an exponentialincrease in computation speed, a secure means of communication and an exponential saving in transmitted information. The efficient distribution of quantum state and entanglement is a key ingredient for such a global platform. Entanglementdistribution and quantum teleportation can be employed to transmit quantum states over long distances. By exploitingentanglement swapping and quantum purification, the transmission distance could be extended significantly and the fidelitiesof transmitted states can be enhanced up to unity, which is known as quantum repeaters. However, with the increased ofcomplexity of quantum networks, especially when many parties require simultaneous communication and communication ratesexceed the capacity of quantum channels, low transmission rates or long delays, known as bottleneck problems, are expected tooccur.Thus, it is important to resolve the bottleneck problem and achieve high-speed quantum communication. This question isin the line with issues related to quantum communication complexity, which attempts to reduce the amount of information to betransmitted to solve distributed computational tasks. The bottleneck problem is common in classical networks. A landmark solution is the network coding concept, wherethe key idea is to allow coding and replication of information locally at any intermediate node in the network. The metadataarriving from two or more sources at intermediate nodes can be combined into a single packet, and this distribution methodcan increase the effective capacity of a network by minimizing the number and severity of bottlenecks. The improvement ismost 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. A 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 dueto the no-cloning theorem. However, remarkable theoretical effort has been directed at this important question. For example,Hayashi et al. were the first to study QNC, and they proved that perfect quantum state cross-transmission is impossible inthe butterfly network, i.e., the fidelity of crossly transmitted quantum states cannot reach one. However, if two senders haveshare entanglements priorly, the perfect QNC is possible by exploiting quantum teleportation. Thus, various studies havefocused on network coding for quantum networks, such as the multicast problem,
QNC based on quantum repeaters, a r X i v : . [ qu a n t - ph ] O c t S C C R R b b b b b b b b b b b b b b BS M BS M Z XZ X Z XZ X S S R R C C + F + F r r r r a b Figure 1. Classical network coding and quantum network coding on a butterfly network. a, classicalnetwork coding on a butterfly network. Dash line with arrow represents information flow with a capacity of a singlepacket. In the two simultaneous unicast connections problem, one packet b presented at source node S isrequired to transmit to node R and the other packet b presented at source node S is required to transmit to node R simultaneously. The intermediate node C performs a coding operation XOR ⊕ on b and b . C makes copies of b ⊕ b and sends them to R and R respectively. R and R decode by performing further XOR operations on thepackets that they each receive. b, quantum network coding on butterfly network. The red line with arrow representsquantum information flow with a capacity of a single qubit, and the dash line with arrow represents classicalinformation flow with a capacity of a two bits. See main text for more details. QNC based quantum computation and other efficient quantum-communication protocols with entanglement. Despitethese theoretical advances, to the best of our knowledge, an experimental demonstration of QNC has not been realized in alaboratory, even for the simplest of cases.In this study, we provide the first experimental demonstration of a perfect QNC protocol on the butterfly network. Thisexperiment adopted the protocol proposed by Hayashi, who proved that perfect QNC is achievable if the two senders havetwo prior maximally-entangled pairs, while it is impossible without prior entanglement. We demonstrate this protocol byemploying eight photons generated via spontaneous parametric down-conversion (SPDC). We observed a cross-transmission ofsingle-photon states with an average fidelity of 0 . ± . . ± . Results
QNC on butterfly network
Network coding refers to coding at a node in a network. The most famous example of network coding is the butterflynetwork, which is illustrated in Fig. 1a. While network coding has been generally considered for multicast in a network, itsthroughput advantages are not limited to multicast. We focus on a simple modification of the butterfly network that facilitatesan example involving two simultaneous unicast connections. This is also known as : which seeks to answerthe following: for two sender-receiver pairs ( S - R and S - R ), is there a way to send two messages between the two pairssimultaneously? 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 present at sender S that we want to transmit to receiver R and a single packet b present at sender S thatwe want to transmit to receiver R simultaneously. The intermediate node C breaks from the traditional routing paradigm ofpacket networks, where intermediate nodes are only permitted to make copies of received packets for output. Intermediatenode C 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 ⊕ b . Ultimately, R recovers b by taking the XOR of b and b ⊕ b and similarly R recovers b by taking the XOR of b and b ⊕ b . Therefore, two unicast connections can be established withcoding and cannot without coding.In the case of quantum 2-pairs problem, the model is the same butterfly network (Fig. 1a) with unit-capacity quantumchannels and the goal is to send two unknown qubits crossly, i.e., to send ρ from S to R and ρ from S to R simultaneously.However, two rules prevent applying classical network coding directly in the quantum case: (i) an XOR operation for twoquantum states is not possible; (ii) an unknown quantum state cannot be cloned exactly. Therefore, it has been proven that thequantum 2-pairs problem is impossible. ayashi proposed a protocol that addresses the quantum 2-pairs problem by exploition prior entanglements between twosenders. As shown in Fig. 1b, the scheme is a resource-efficient protocol that only requires two pre-shared pairs of maximallyentangled state | Φ + (cid:105) between the two senders. Notice that if the sender ( S , S ) nodes and receiver ( R , R ) nodes allow to S U C C R R m n m n m n m n m n m n m n m n S U U U
12 3487 56 BSM BSM U i i m ni X Z = X Z
X Z
1 0
X Z
X Z ac BBOPrismPBSMirroHWPQWPSPD
FPGA
CopyXOR d FPGA
Sender1Sender2 Reciver2Reciver1 r r r¢ r¢ b o-raye-ray Figure 2. Schematic drawing of the experimental setup. a, an ultraviolet pulse successively pass through fourBBO crystals, and generate four pairs of maximally entangled photons. We use four Bell-state synthesizer (shownin Fig.2b). to improve the counter rate of entangled photon pair. To avoiding a mess of illustration, we separatedpropagation of ultraviolet pulse. In our experiment, the ultraviolet pulse is guided by mirrors to shine on four BBOone by one. All the photons are collected by single-mode fiber and detected by single photon detecters (SPD). Thecoincidence is recored by several home-made field-programmable gate arrays (FPGA). See main text for moredetails. b, Bell-state synthesizer. The generated photons are compensated by a HWP at 45 ◦ and 1-mm-long BBOcrystal. Then, one photon is rotated by a HWP at 45 ◦ and finally two photons are recombined on a PBS. WithBell-state synthesizer makes ordinary ray(o-ray) exiting from one port of PBS and extraordinary ray(e-ray) exitingthe 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 onlyby using quantum teleportation. If free classical between all nodes is not limited, perfect 2-pair communication over thebutterfly network is possible. 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. Hayashiproved that the average fidelity of quantum state transmitted is upper bounded by 0.9504 for single-qubit state and 0.9256 forentanglement without prior entanglement. However, with prior entanglement between senders, the average fidelity can reach1. The protocol is summarized as follows (see Fig. 1b).1. S ( S ) applies the Bell-state measurement (BSM) between the transmitted state ρ ( ρ ) and one qubit of | Φ + (cid:105) . Accordingto the result of BSM m n ( m n ), S ( S ) perform the unitary operation X m Z n ( X m Z n ) on the other qubit of | Φ + (cid:105) .2. S ( S ) sends the quantum state (after the unitary operation) to R ( R ), and sends the classical bits m n ( m n ) to C . C performs the XOR on m and m , n and n respectively, then sends m = m ⊕ m and n = n ⊕ n to C . C makes H HV H+ H- HR HL VH VV V+ V- VR VL +H +V ++ +- +R +L -H -V -+ -- -R -L RH RV R+ R- RR RL LH LV L+ L- LR LL
Node S to Node R Node S to Node R State combinations S t a t e F i de li t y a b State fidelity P r obab ili t y Figure 3. Fidelities of crossly transmitted quantum states. a,
The green bar represents F S → R , and the yellowbars represents F S → R . The pair-appeared bars represent fidelities measured simultaneously at R and R . Forexample, HR means that S delivers | H (cid:105) and S delivers | R (cid:105) . The red line represents the threshold of F t h = . .The fourfold coincidence is approximately 1.5 counts per second. We accumulate coincidences for 240 secondsand a total of 720 counts for each two-state transition. The error bars are calculated assuming a Poisson statisticsfor the coincidence counts and Gaussian error propagation. b, Histogram of state fidelities. copies of m n and sends them to R and R , respectively.3. R and R recover the quantum states ρ and ρ by applying the unitary operation X m Z n on their received quantumstates. Experimental realization
We demonstrate the perfect QNC protocol by employing the polarization degree of freedom of photons generated via SPDC. Asshown in Fig. 2a, an ultraviolet pulse (with a central wavelength of 390 nm, power of 100 mW, pulse duration of 130 fs andrepetition of 80 MHz) successively passes through four 2-mm-long BBO crystals successively and generates four maximallyentangled photon pairs via SPDC in the form of | Ψ + (cid:105) i j = √ ( | HV (cid:105) + | V H (cid:105) ) i j . 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 betweentwo photons (as shown in Fig. 2b). After the Bell-state synthesizer, | Ψ + (cid:105) i j is converted to | Φ + (cid:105) i j = √ ( | HH (cid:105) + | VV (cid:105) ) i j .We set narrow-band filters with full-width at half maximum ( λ FWHM ) 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 visibilityof 99.6% in the | H ( V ) (cid:105) basis and visibility of 99.0% in the | +( − ) (cid:105) basis, from which we calculate the fidelity of preparedentangled photons with an ideal | Φ + (cid:105) of 99.3%. We estimate a single-pair generation rate of p ≈ . | Φ + (cid:105) and | Φ + (cid:105) are the two entangled pairs priorly shared between S and S , i.e., S holds photon 1&3 and S holdsphoton 2&4. | Φ + (cid:105) and | Φ + (cid:105) are held by S and S , respectively. Photon 5 is projected on α ∗ | H (cid:105) + β ∗ | V (cid:105) to prepare ρ withideal form in α | H (cid:105) + β | V (cid:105) . Similarly, photon 7 is projected on α ∗ | H (cid:105) + β ∗ | V (cid:105) to prepare ρ with ideal form α | H (cid:105) + β | V (cid:105) .On S ’s side, by finely adjusting the position of the prism on the path of photon 1, we interfere with photons 1 and photon6 on a polarizing beam splitter (PBS) to realize a Bell-state measurement (BSM). The BSM projects photons 1 and photon6 to | ψ (cid:105) ∈ {| Ψ + (cid:105) , | Ψ − (cid:105) , | Φ + (cid:105) , | Φ − (cid:105)} . As the complete BSM is impossible with linear optics, we perform the completemeasurements with two setup settings by rotating the angle of the half-wave plate (HWP) on path 6 or 1 before they interferefrom 0 ◦ to 45 ◦ . Note that in each setup, the success probability to identify two of the Bell states is 50%. So, the total successprobability is 25% in our experiment. The BSM results (different responses on the four detectors after the interference) arerelated to two classical bits denoted as m n ∈ { , , , } . According to the BSM results, S applies the unitary operation U = X m Z n on photon 3, and then sends m n to node C and photon 3 to the receiver node R . Here, we use X , Y , Z torepresent the Pauli- X , Pauli- Y , Pauli- Z matrix. Similarly, on the S side, we interfere with photons 4 and 8 on a PBS to realize aBSM with result of m n , according to which S applies the unitary operation U = X m Z n on photon 2. Then, S sends m n to node C and sends photon 2 to the receiver node R .On node C , we perform the XOR operation on m and m and n and n , and send the results m = m ⊕ m , n = n ⊕ n to node C , where we make two copies of m n and send these copies to R and R . Finally, according to m n , we apply theunitary operation U = X m Z n on photons 3 and photon 2 to recover ρ and ρ . Node S to Node R Node S to Node R + + + - + + + - - + - - - + - - ++ -+ + + + - +- -- - + - - S t a t e F i de li t y Choices of two BSMs
Figure 4. Fidelities of crossly established entanglement.
The green bar represents entanglement establishedbetween S to R , and the yellow bars represents S to R . The pair-appeared bars represents the fidelitiesmeasured simultaneously. For each BSM, there are four possible outcomes, thus there are 16 situations. The redline represents the threshold of F th = . . The fourfold coincidence is about 3 counts per second. Weaccumulate coincidences for 120 seconds and total 720 counts for each two-state transition. The error bars arecalculated assuming a Poisson statistics for the coincidence counts and Gaussian error propagation. In our experiment, the unitary operation is realized by HWPs with transformation matrix U ( θ ) = (cid:18) cos θ sin θ sin θ − cos θ (cid:19) ,where θ is the angle fast axis relative to the vertical axis. X Z = I means no operation on the photon. Here, X Z = Z isrealized by setting an HWP at 0 ◦ . X Z = X is realized by setting an HWP at 45 ◦ , and X Z = XZ is realized by setting twoHWPs (one at 45 ◦ and the other at 0 ◦ (Shown in Fig.2c)) Experimental results
We first show that two single-photon states can be crossly delivered from S to R and from S to R simultaneously in thebutterfly network. S and S can prepare six individual quantum states ρ and ρ with an average fidelity of 99.3%. ρ and ρ have an ideal form of ρ = | φ (cid:105) (cid:104) φ | and ρ = | φ (cid:105) (cid:104) φ | , where | φ (cid:105) , | φ (cid:105) ∈ {| H (cid:105) , | V (cid:105) , |±(cid:105) = √ ( | H (cid:105) ± | V (cid:105) ) , | L ( R ) (cid:105) = √ ( | H (cid:105) ± i | V (cid:105) ) } . In our experiment, both S and S irrelatively select ρ and ρ from six states for transmission, therebyresulting in a total of 36 combinations. After recover of R and R , we measure the fidelities between the recovered state ρ (cid:48) ( ρ (cid:48) ) and the ideal input state ρ = | φ (cid:105) (cid:104) φ | ( ρ = | φ (cid:105) (cid:104) φ | ), i.e., F S → R = Tr ( | φ (cid:105) (cid:104) φ | ρ (cid:48) ) and F S → R = Tr ( | φ (cid:105) (cid:104) φ | ρ (cid:48) ) . Weproject the photon on the | φ (cid:105) ( | φ ⊥ (cid:105) ) basis and record the counts N + and N − , where | φ ⊥ (cid:105) is the orthogonal state of | φ (cid:105) . Thus,the fidelity of the transferred single-photon state can be calculated by F = N + N + + N − . The average fidelities over all possibleBSM outcomes are shown in Fig. 3a. Note that each BSM has four possible outcomes, thus there are 16 combinations ofoutcomes for the two BSMs. For each combination, we apply the unitary operations and record the measured fidelities. InFig. 3a, the red line represents the theoretical upper bound of the average fidelity without prior entanglement, i.e., F th = . F = ∑ i p i F i = . ± . p i and F i are the probability and fidelity shown in Fig. 3b. The average fidelity beyonds F th = . S and R and S and R simultaneously. Here, the experimental setup is the same, S ( S ) does notproject photon 5(7) on α ∗ | H (cid:105) + β ∗ | V (cid:105) , and photon 5(7) is retained to perform the joint measurements with photon 2(3). Toquantify the cross entanglement between photon 5 and 2 and 7 and 3, we measure the entanglement witness on rho and ρ , respectively. In particular, we measure the entanglement witness W = I / − | Φ + (cid:105) (cid:104) Φ + | , which can also be related to theentanglement fidelity (cid:104) W (cid:105) = / − F ent . Here F ent is defined as the entanglement fidelity between the entanglement state ρ i j andthe maximal entanglement state | Φ + (cid:105) , i.e., F ent = Tr ( ρ i j | Φ + (cid:105) (cid:104) Φ + | ) . | Φ + (cid:105) (cid:104) Φ + | can be decomposed to a local observable as | Φ + (cid:105) (cid:104) Φ + | = II + XX − YY + ZZ . By measuring the expected values of local observables, we can calculate the entanglement fidelity.The local observable O can be expressed as O = | φ (cid:105) (cid:104) φ | − | φ ⊥ (cid:105) (cid:104) φ ⊥ | , where | φ (cid:105) ( | φ ⊥ (cid:105) ) is the eigenstate of O with eigenvalue of1(-1). The expected value of O can be calculated by the counts (cid:104) O (cid:105) = N + − N − N + + N − . The experimental results of the fidelities of cross ntanglement are shown in Fig. 4. We calculate that the average fidelity of two crossly established is 0 . ± . Discussion
QNC provides an alternative solution for the transition of quantum states in quantum networks. Compared to entanglementswapping, QNC demonstrates superiority especially when quantum resources are limited or a high communication rate isrequired. In addition, large-scale QNC demonstrates superiority relative to fidelity-performance as well. In this paper, Wehave demonstrated the first perfect QNC on a butterfly network. The average fidelities of cross state 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 quantumcommunication. In addition, we expect that our results will realize opportunities for various studies of efficient quantumcommunication protocols in quantum networks with complex topologies.
Author contributions
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 withcontributions from all authors.
Competing interests
The authors declare no competing interests.
Data availability
The data are available from the corresponding author upon reasonable request.
Acknowledgement
This work was supported by the National Key Research and Development (R&D) Plan of China (grants 2018YFB0504300 and2018YFA0306501), the National Natural Science Foundation of China (grants 11425417, 61771443), the Anhui Initiative inQuantum Information Technologies and the Chinese Academy of Sciences. H. Lu was partially supported by Major Program ofShandong Province Natural Science Foundation (grants ZR2018ZB0649). F. Xu thanks Prof. Bin Li for the early inspiration tothe subject.
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