Abstract
Multipartite entangled states are important for developing studies of quantum networking and quantum computation. To date, the largest number of particles that have been successfully manipulated is 14 trapped ions. Yet in quantum information science, photons have particular advantages over other systems. In particular, they are more easily transportable qubits and are more robust against decoherence. Thus far, the largest number of photons to have been successfully manipulated in an experiment is six. Here we demonstrate, for the first time, an eightphoton Greenberger–Horne–Zeilinger state with a measured fidelity of 0.59±0.02, which proved the presence of genuine eightpartite entanglement. This is achieved by improving the photon detection efficiency to 25% with a 300mW pump laser. With this state, we also demonstrate an eightparty quantum communication complexity scenario. This eightphoton entangledstate source may be useful in oneway quantum computation, quantum networks and other quantum information processing tasks.
Introduction
The creation and coherent manipulation of multiphoton entangled states has a crucial role in photonic quantuminformation processing. In most cases, a pulsepumped spontaneous parametric downconversion (SPDC) process is used for the generation of multiphoton entangled states in which n photon pairs from the SPDC's nthorder emission are available. In particular, polarizationentangled Einstein–Podolsky–Rosen (EPR) pairs are often used as the basic blocks for generating multiphoton entangled states or as resources for performing quantum teleportation tasks. A number of experiments involving two photon pairs^{1,2,3,4,5,6,7,8} and a few experiments involving five photons (two photon pairs plus one single photon)^{9} or three photon pairs^{10,11,12,13} have been carried out to demonstrate various protocols in quantum communication and linear optical quantum computing. Typical systems for twophotonpair experiments use a frequencydoubled Ti:sapphire modelocked laser as a pump, with a repetition rate of ~80 MHz, a pulse duration of ~150 fs, and an operating wavelength of ~400 nm. The improvement in moving from twophotonpair to threephotonpair experiments depends on the pump power being increased from 600 mW to 1 W through the use of a highpower continuouswave laser to pump the modelocked laser^{14}. However, the sixfold coincidence count rates under this pump power are still low, and it is difficult to increase the pump power further in order to manipulate more photon pairs.
In fact, other methods have been developed in recent years that are capable of generating more photon pairs under moderate pump power, such as sources obtained using spontaneous processes in waveguide structures^{15} or photonic crystal fibres^{16,17}. More recently, an ultraviolet enhancement cavity suitable for multiphoton experiments was implemented by Roland Krischek et al.^{18} This system can provide greater than 7 W ultraviolet pump power for SPDC processes, which also makes the generation of more photon pairs possible. However, because of the nature of spontaneous processes, the probability μ of generating one photon pair per pump pulse should normally be restricted to less than 0.1 such that the noise from higher order photonpair emission can be neglected. For an experiment involving n photon pairs from a spontaneous process, the 2nfold coincidence counting rate can be roughly estimated as C_{2n}=fμ^{n}η^{2n}, where f is the pump laser repetition rate and η is the total detection efficiency, which is the product of all efficiencies between the source and detector. Thus, under the limit of μ≤0.1, a key parameter for a good multiphoton source is the value of η, which significantly affects C_{2n}, especially for large n.
In this study, for the first time, we demonstrate an experiment involving genuine eightphoton entanglement, which is achieved by improving η to 0.25 and setting μ to be 0.046. With this source, we demonstrate an eightparty quantum communication complexity scenario using an eightphoton Greenberger–Horne–Zeilinger (GHZ) state with a fidelity of F=0.59±0.02. Our experimental result has a success probability largely exceeding the classical limit of 56.25%, which shows that the eightphoton entangled state we prepared violates the Mermin inequality^{19}.
Results
Theoretical schemes
Communication complexity problems (CCPs)^{20} describe the communication cost incurred in performing distributed computation tasks among a number of separated parties, with each party holding his or her own input data at the beginning. As the quantum mechanical counterpart of CCPs, quantum communication complexity problems (QCCPs) involve the same problems, but the different parties communicate by sending quantum bits instead of classical bits^{21} (or by sending classical bits through previously shared entanglements^{22}). It has been shown^{23,24} that QCCPs can have quadratic or even exponential superiority over classical communication complexity in some cases. It has also been proven that a violation of Bell inequalities is the necessary and sufficient condition for QCCPs to outperform classical CCPs^{25}.
The special quantum communication complexity scenario (QCCS) that we demonstrate in our experiment is an eightparty version of Buhrman's model^{26}. Suppose that there are eight parties A1, A2,...,A8 receiving data X_{1}, X_{2},...,X_{8}, respectively, where X_{i}U{0, 1}^{2}, i=1, 2,...,8, and
The common objective of each party is to obtain the correct value of the Boolean function
X_{1},...,X_{8} can each be written in binary notation as x1_{1}x1_{0},..., x8_{1}x8_{0}. Under the conditions of equation (1), for any input X_{1},...,X_{8}, the value distributions of x1_{0},...,x8_{0} should be one of 128 possible combinations, which can be divided into five groups, namely, 0^{8}, 0^{6}1^{2}, 0^{4}1^{4}, 0^{2}1^{6} and 1^{8}. Here 0^{i}1^{j} stands for a combination that has i bits of 0 and j bits of 1, regardless of the order in which they appear. The Boolean function F can be rewritten as
where
That is, F_{0}=0 for the groups 0^{8}, 0^{4}1^{4} and 1^{8}, whereas F_{0}=1 for 0^{6}1^{2} and 0^{2}1^{6}.
In the classical situation, the maximum success probability for this scenario is
where N is the number of parties. In our case, N=8 so that . Using quantum protocols, the eight parties X_{1}−X_{8} that share a prior eightqubit GHZ state can have a success probability of . The protocol is shown below.
The eightqubit GHZ state is
Each party holds a qubit of and applies rotations R(x) or R(y) to the qubit according to the value (that is, 0 or 1) of its own classical bit xi_{0}, i∈1, 2...8, where
Next, the party measures its qubit in the 0/1 basis and obtains the result yi∈{0, 1}. After that, each party X_{i} broadcasts a bit and can derive the correct value of F as
with a success probability of 100%.
Experimental realization of eightphoton GHZ state and QCCS
We now proceed with the experimental description of how to generate the eightphoton GHZ state and demonstrate the QCCS scheme. Four specially designed ultrabright 'beamlike' EPR sources (S1 to S4)^{27} were used in the experiment to provide polarizationentangled EPR photon pairs. The experimental setup is illustrated in Figure 1. A pulsed ultraviolet laser beam was split into two beams by a 50/50 nonpolarizing beamsplitter, and each beam was subsequently passed through two pairs of βbarium borate (BBO) crystals serving as the pump. Finally, 4 EPR pairs were generated in 8 different spatial modes, numbered from 1 to 8. The reason for using two pump beams, each generating two EPR pairs, instead of using one pump beam generating four EPR pairs, was to reduce the number of lenses that the pump beam passed through before pumping the latter two EPR sources (S3 and S4). Passing through too many lenses would lead to a large decrease in the downconverted photon detection efficiency, for example, a decrease from 0.23 to 0.21. Furthermore, all of the lenses were optimized according to their focal lengths to obtain the highest collection efficiency η and a moderate production efficiency μ. The focal lengths of L1, L2, L3 and L4 were 150 mm, 85 mm, 150 mm, and 85 mm, respectively, and the distances between S1 and S2 and between S3 and S4 were both 300 mm.
To achieve the best stability of the system, eight polarizationpreserving singlemode fibres were placed as close as possible to the BBO crystals to collect the downconverted photons in the eight spatial modes. In addition, we also constructed a new type of frequency doubler to improve the total detection efficiency η (Methods), which significantly increases the eightphoton coincidence rate without increasing the generation probability μ, thus keeping the noise of higher order photonpairemission from increasing. With these techniques and using only 300 mW pump power for each 'beamlike' EPR source, we observed ~22×10^{4} s^{−1} photon pairs from each EPR source.
The photon pairs were prepared according to the Bell state , where H and V denote horizontal and vertical polarization, respectively. The visibility with respect to the H/V (+/−) basis under 300 mW pump power was measured to be 96% (93%) where . The outputs of these fibres were directed to polarization beamsplitters (PBS) to perform postselected fusion operations^{28} between modes 1 and 3 (PBS1), and 6 and 7 (PBS2) or polarizationprojection measurements for modes 1, 4, 5 and 8. Next, the transmitted modes of mode 3 and 7 after PBS1 and PBS2 were directed to PBS3 for another fusion operation. An eightphoton GHZ state was prepared by postselecting one and only one photon in each of the eight modes labelled 1′, 2′, 3′, 4′, 5′, 6′, 7′ and 8′. Next, in each mode, a quarter waveplate (QWP) and halfwaveplate (HWP) were used to perform the R(x) or R(y) rotations (depending on the value of xi_{0}), and a PBS was placed after the HWP to derive the H/V basis measurement, where a result of H› denotes 0 and a result of V› denotes 1. Finally, the photons were spectrally filtered (Δλ_{FWHM}=3 nm) for good temporal indistinguishability and detected by spacecoupled singlephoton detectors for better collection efficiency. Note that the spatial modes were filtered by previous single mode fibres. Two singlephoton detectors were employed for each photon, with one to detect H polarization and the other to detect V polarization. A homemade, programmable, 16channel coincidence unit was used to simultaneously register all possible coincidences among the 16 singlephoton detectors.
Because of the long data collection period in our experiment (~60 h for one measurement basis) and strong focus of the ultraviolet laser beam on downconversion BBO crystals, we also proposed an antiburning method to avoid damage to the BBO crystals caused by the laser beam. For this method, each EPR pair source was mounted on a translation stage that moved up and down repeatedly in a small region to reduce heating due to the strongly focused ultraviolet pulses. This antiburning system provided good stability to the pump power after it passed through the BBO crystals. Using this method, the stable time of the pump power (defined as the variation of pump power being less than ±2%) greatly increased from less than 2 h to months. It is also worth noting that in our experiment, the interference of two independent photons occurred between the outputs of two singlemode fibres mounted on stable translational stages. This adaptation greatly improved the stability of our interferometer. Under 300 mW pump power for each SPDC source, the average visibility of Hong–Ou–Mandel (HOM) interferences between independent photons overlapped on PBS1, PBS2 and PBS3 was observed to be ~73%.
We can verify genuine eightphoton entanglement of our prepared eightphoton GHZ state using the witness^{29}
where
and
The state fidelity can then be directly calculated as
Figure 2a shows the eightfold coincidence counts of our prepared eightphoton GHZ state measured in H/V basis. Approximately 80 eightphoton events were observed in 64 h for the two desired columns of HHHHHHHH and VVVVVVVV, while undesired events in the other 254 columns clearly showed effects from the higher order photonpair emissions in our experiments. The effect from PBS imperfections was small, because every PBS in our system had an extinction ratio of more than 300:1 for both output ports. From Figure 2a, the value of observable was calculated to be 0.75±0.03. Figure 2b illustrates the expectation values of M_{k}, k=1, ..., 8. Each expectation value was derived from a complete set of eightphoton coincidence counts (256 kinds) obtained by measuring each photon on the basis of . We obtained an averaged expectation value of (−1)^{k}M_{k} of from Figure 2b. Therefore, we can evaluate the expectation value of W_{G} as ‹W_{G}›=−0.09±0.02, which clearly shows the presence of genuine eightphoton entanglement. Therefore, the fidelity of our prepared GHZ state was according to equation (11).
To demonstrate the eightparty QCCS, there were 128 kinds of possible inputs of X_{1}, ..., X_{8}, and eight communication parties were expected to perform different local rotations on their qubits for different inputs. Measuring all 128 possible cases was difficult, because of the long datacollection period of ~50 h for each case. The 128 possible cases were thus divided into five groups denoted 0^{8}, 0^{6}1^{2}, 0^{4}1^{4}, 0^{2}1^{6} and 1^{8}, as mentioned above. Therefore, in our experiment, we only studied one case in each group (that is, 00000000, 11000000, 11110000, 11111100, and 11111111). Because of the symmetry of our stategeneration setup, it is reasonable to assume that exchanging the places of 1 and 0 among the eight parties would yield no differences in the experimental results. Figure 3 shows the measured success and fail probabilities of the five cases. In each group, the success probability P_{s} largely exceeded the classical limit of 56.25%. In addition, the average success probability Pa was 71.7±1.4%. In other words, the eightphoton state we prepared cannot be described by local realistic theory.
Discussion
The imperfections in our experiment had two main causes. One cause involved the distinguishability between two independent photons and an imperfect EPR source, which were characterized by the HOM interference visibility V_{HOM} of independent photons and the entanglement visibility V_{Ent} of the EPR source measured under low pump power. In our experiment, V_{HOM} and V_{Ent} were ~82% and 97%, respectively. The other important factor was the higher order photonpair emissions due to the nature of SPDC processes. The effect of this factor can be directly observed from the 256 measured eightphoton counts in the H/V basis.
In summary, we have presented a novel source that can provide four independent EPR photon pairs for multiphoton experiments using only 300 mW pump power for each SPDC source. The experimental setup of this source is rather simple, stable and easy to manipulate. Using four EPR photon pairs and three fusion operations, we successfully generated an eightphoton GHZ state with verified genuine eightphoton entanglement. With this GHZ state, we demonstrated an eightparty quantum communication complexity scenario in an experiment, with the result exceeding the classical limit of the eightparty communication complexity scenario. A number of further applications can be imagined with our eightphoton entanglement source; for instance, the demonstration of various protocols in multiparty quantum networking^{12,30}, losstolerant oneway quantum computing using cluster states^{31}, highresolution quantum optical metrology with multiphoton states^{32}, and tests of Bell's theorem without inequalities and alignments^{33}. In recent years, the planar integrated optics approach of optical quantum circuits has shown tremendous potential for future optical quantum information processing, such as Shor's algorithm being demonstrated on a chip with four single photons^{34}. The integrated optics approach has the advantages of perfect spatial overlap and a compact and stable setup compared with traditional tableoptics systems. Therefore, the possible combination of our highbrightness, highcollectionefficiency SPDC source and planar integrated optics might bring further advancement to optical quantum information technology in the future.
Methods
Frequency doubler for high photoncollection efficiency
The collection efficiency of the photon pairs from the SPDC source was an important factor in reducing the data collection time in our eightphoton experiment. On the basis of our previous beamlike EPR source^{27}, we created a new type of frequency doubler that remarkably improved the photonpair collection efficiency. The experimental setup of our frequency doubler is shown in the inset of Figure 1. In our frequency doubler, we used a BiB_{3}O_{6} (BIBO) crystal instead of a LiB_{3}O_{5} (LBO) or BBO crystal, and the BIBO crystal was cut at θ=149.5° and φ=90° to perform a type I secondharmonic generation (SHG) process. We found that the spectral bandwidth of the output ultraviolet beam from our frequency doubler was narrower than other frequency doublers that use LBO or BBO crystals. In our case, the spectral bandwidth of the 390 nm laser beam was 0.9 nm (full width of halfmaximum (FWHM)). Such a narrow bandwidth of pump significantly improves the downconverted photonpair collection efficiency. We also found that to achieve the best collection efficiency of photon pairs, the value of the BIBO's phasematching angle θ should be slightly different from the value of the highest SHG output power. Another improvement of the collection efficiency was achieved by ensuring that the focusing in the frequency doubler was not too strong. We chose a focal length of 125 mm for the focusing lens before the BIBO crystal to obtain the best collection efficiency while maintaining enough SHG output power.
With the abovementioned techniques and other arrangements for improving photon detection efficiency described in the main text, such as splitting the 390nm pump beam into two beams and choosing optimized focal lengths for pumpbeam focusing, we improved the average total detection efficiency η from the original value of 0.125 to 0.250. This result increased the eightphoton coincidence counting rate by over 200 times under the same generation probability μ, thereby making the eightphoton entanglement experiment feasible.
It is important to note that because of the narrower spectral bandwidth of the 390nm laser beam, the visibility of HOM interference between independent photons in our system decreased. In fact, when using the low power of a 390nm laser beam (~50 mW) to remove the emission background of the higher order photon pairs, we observed an average HOM interference visibility of ~83% and ~90% when BIBO and BBO crystals were used in the frequency doubler, respectively.
Additional information
How to cite this article: Huang, Y.F. et al. Experimental generation of an eightphoton Greenberger–Horne–Zeilinger state. Nat. Commun. 2:546 doi: 10.1038/ncomms1556 (2011).
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Acknowledgements
We thank YanXiao Gong for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos 10874162, 11074242, 11104261 and 60921091), the National Basic Research Program of China (Grant Nos 2011CB921200 and 2011CBA00200), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant No. 200729), the Fundamental Research Funds for the Central Universities (Grant Nos WK2030020004 and WK2030020007), and the Anhui Provincial Natural Science Foundation (Grant No. 11040606Q47).
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C.F.L. and Y.F.H. designed the experiment. Y.F.H., B.H.L. and L.P. performed the experiment. Y.H.L. made the 16channel coincidence unit. Y.F.H. and C.F.L. supervised the project. L.L. and G.C.G. contributed to the theoretical analysis, Y.F.H. analysed the data and wrote the paper.
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Huang, YF., Liu, BH., Peng, L. et al. Experimental generation of an eightphoton Greenberger–Horne–Zeilinger state. Nat Commun 2, 546 (2011). https://doi.org/10.1038/ncomms1556
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DOI: https://doi.org/10.1038/ncomms1556
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