Abstract
Graph states^{1,2,3}—multipartite entangled states that can be represented by mathematical graphs—are important resources for quantum computation^{4}, quantum error correction^{3}, studies of multiparticle entanglement^{1} and fundamental tests of nonlocality^{5,6,7} and decoherence^{8}. Here, we demonstrate the experimental entanglement of six photons and engineering of multiqubit graph states^{9,10,11}. We have created two important examples of graph states, a sixphoton Greenberger–Horne–Zeilinger state^{5}, the largest photonic Schrödinger cat so far, and a sixphoton cluster state^{2}, a stateoftheart ‘oneway quantum computer’^{4}. With small modifications, our method allows us, in principle, to create various further graph states, and therefore could open the way to experimental tests of, for example, quantum algorithms^{4,12} or loss and faulttolerant oneway quantum computation^{13,14}.
Main
Entanglement lies at the heart of quantum mechanics and plays a crucial role in quantuminformation processing. Many efforts have been undertaken to create, in particular, multipartite entangled states in different physical systems^{15,16,17,18}, where maximally up to eight ions have been entangled^{18}. In recent years, special types of multipartite entangled states, the graph states^{1,2,3}, have become the centre of attention. They can be associated with graphs where each vertex represents a qubit prepared in the state and each edge represents a controlled phase gate having been applied between the two connected qubits. An interesting feature is that many entanglement properties of graph states are closely related to their underlying graphs^{1}. Besides their thoughtprovoking theoretical structure, the graph states have also provided new insights into studies of nonlocality^{6,7,19,20} and decoherence^{8} and served as essential resources for various quantuminformation tasks^{3,4}, most prominently as the exceptionally universal resource for oneway quantum computation^{4}. Encouraging progress^{9,10,11,12,13,14,16} has been achieved in this direction, especially in the linear optics regime. Yet a major challenge ahead lies in the experimental generation of multiqubit graph states.
Of special interest in the graphstate family are the Greenberger–Horne–Zeilinger (GHZ) states and the cluster states. Experimentally, sixatom GHZ states^{17} and fourphoton cluster states^{16} have been realized. Here, we report the creation of sixphoton GHZ states and cluster states with verifiable sixpartite entanglement. To do so, we start from three Einstein–Podolsky–Rosen (EPR) entangled photon pairs in the state
where H and V denote horizontal and vertical polarization, and i and j label the spatial modes of the photons (see Fig. 1a). We superpose photons in mode 2 and 3 (4 and 5) at polarizing beam splitters (PBSs). As the PBS transmits H and reflects V polarization, only if both incoming photons have the same polarization can they go to different outputs^{21,22}. Thus, a coincidence detection of all six outputs corresponds to the state
which is a sixphoton GHZ state, exhibiting an equal superposition of two maximally different quantum states.
By applying a Hadamard gate on photon 4 before it enters into the PBS (see Fig. 1a), the above scheme can be readily modified to generate a sixphoton cluster state. It can be considered in two steps: (1) combine photons 2 and 3, such that, on the basis of a coincidence detection, we get a fourphoton GHZ state , where ; (2) combine photons 4 and 5, and by a similar reasoning we obtain what we call here a sixphoton cluster state
For an intuitive understanding, in Fig. 1 we show the underlying graph of the above states and how they grow from smaller (twoqubit) graph states. Up to local unitary transformations, the GHZ states correspond to starshaped graphs, and the cluster states to lattice graphs. The effect of combining two photons at PBS can be described by the operator H H〉〈H H+V V 〉〈V V , leading to the fusion of two separate graph states into a single one^{10,11}. Specifically, Fig. 1c (d) shows that when a twoqubit graph state is combined with the root (leaf) node of a fourqubit star graph, a sixqubit GHZ (cluster) state is produced.
A nice feature of the graphstate representation is that many properties of the graph states and their potential use in quantuminformation processing can be revealed by their underlying graph. For example, the stargraph states have multiple leaf nodes, which are referred to as microclusters in refs 913 and can be used in the socalled parallel fusion for building up large cluster states. The graph of the sixqubit cluster state (2) forms a standard quantum circuit under the oneway computer model^{4}. Moreover, its geometry embodies a treeshaped graph, which is the basic building block for losstolerant oneway quantum computing^{14}. Another interesting feature of the cluster state is that not only itself, but even the remaining mixed fourqubit state after two qubits have been traced out, leads to a GHZ argument for nonlocality^{6}, showing a surprisingly strong entanglement persistency.
Let us now proceed with the experimental demonstration. In our experiment, we use spontaneous parametric downconversion to produce entangled photons^{23}. We made various efforts to prepare highbrightness and stable sources of entangled photons (see the Methods section). The setup is illustrated in Fig. 2. A pulsed ultraviolet laser successively passes through three βbarium borate (BBO) crystals to generate entangled photon pairs in spatial modes 1–2, 3–4 and 5–6. The photon pairs are prepared in the state Φ^{+}〉 with an average twofold coincidence count of about 9.3×10^{4} s^{−1} and a visibility of 93% (91%) in the H/V (+/−) basis. We then superpose photons 2 (4) and 3 (5) at the PBS. To achieve good spatial and temporal overlap, the photons are spectrally filtered (Δλ_{FWHW}=3.2 nm) and detected by fibrecoupled singlephoton detectors. By making fine adjustments of delay Δd_{1} (Δd_{2}), we are able to observe interference fringes of fourphoton entanglement with a visibility of 73% (71%) in mode 1–2–3–4 (3–4–5–6), indicating that the postselected fusion operations have been successfully implemented (see the Supplementary Information).
Now we analyse the experimental data of sixphoton graph states and characterize the entanglement produced here. Let us first discuss to what extent the desired states were produced and the presence of genuine multipartite entanglement. The quality of the states can be judged by the fidelity, that is, the overlap of the produced state with the desired one. The notion of genuine multipartite entanglement characterizes whether generation of the state requires interaction of all parties: a pure state Ψ〉 is called biseparable, whenever a grouping of the six parties into two groups G_{A} and G_{B} can be found, such that the state is a product state, that is , otherwise it is a genuine multipartite entangled state. Consequently, a mixed state is called biseparable, if it is a mixture of biseparable pure states, otherwise it is a genuine multipartite entangled state.
To prove multipartite entanglement, we use the method of entanglement witnesses^{25}. An entanglement witness is an observable that has a positive expectation value on all biseparable states. Thus, a negative expectation value proves the presence of genuine multipartite entanglement. In what follows, we derive efficient entanglement witnesses that are both robust against realistic noise and economical for experimental efforts.
For the sixphoton GHZ state (1), we use the witness^{25}
where I denotes the identity operator. We decompose G_{6}〉〈G_{6} into locally measurable observables
where M_{(n)}=cos(nπ/6)σ_{x}+sin(nπ/6)σ_{y} are measurements in the x–y plane. To implement this witness, seven measurement settings are required. Figure 3 shows the measurement results, yielding Tr(W_{G}ρ_{exp})=−0.093±0.025, which is negative by 3.7 standard deviations and thus proves the presence of genuine sixpartite entanglement.
From the expectation value of the witness, we can directly determine the obtained fidelity as
F G 6 = 〈 G 6  ρ exp  G 6 〉 =0.593±0.025, where ρ_{exp} denotes the experimentally produced state. This is a considerable improvement over the fidelity of the sixatom GHZ states^{17} (F=0.509±0.004).For the cluster state (2), a possible witness would be W_{C}=I/2−C_{6}〉〈C_{6} (ref. 26). Similar to the constructions of ref. 26, we use a slightly different witness , the implementation of which requires only six measurements (see the Methods section). Figure 4 shows the measurement results in basis , which together with those of the four other bases and (see the Supplementary Information), gives . Thus, the genuine sixpartite entanglement of the cluster state is also proved. Furthermore, from this result, we can obtain a lower bound of the fidelity of our cluster state as F C 6 ≥0.595±0.036.
For an investigation of the bipartite entanglement properties of these graph states, we estimate the entanglement of formation from the expectation value of the witness^{27}. Here, different bipartitions arise when the six parties are divided into two groups. The entanglement of formation E_{F}(ρ) is an entanglement measure for bipartite systems, quantifying how many EPR pairs are needed for the formation of the state^{28}. For the GHZ state (1), we find that for all bipartitions at least E_{F}(ρ_{exp})≥0.073±0.032. For the cluster state (2), E_{F}(ρ_{exp}) is also always positive, for some bipartitions it is even E_{F}(ρ_{exp})≥0.729±0.106. A full discussion, also for a different entanglement measure, is given in the Supplementary Information.
The imperfections of our graph states are mainly caused by two reasons. First, highorder emissions of entangled photons give rise to the undesired components in the H/V basis (see Fig. 3a). Second, the partial distinguishability of independent photons causes some incoherent mixtures. In spite of the imperfections, genuine entanglements of the sixphoton graph states are strictly confirmed. It is possible to improve the fidelity in future experiments, for example, by using photonnumber discriminating detectors to filter out the events of double emissions of photon pairs. Moreover, graph states with high purity can be obtained efficiently using the existing entanglement purification scheme^{29}. The linear optical elements such as the PBS may offer a highaccuracy tool for this task^{30}. It leaves a crucial open question of how to reach the noise thresholds for optical clusterstate quantum computation^{13}.
Some further remarks are warranted here. We generate the graph states conditioned on there being one and only one photon in each of the six outputs. This postselective feature, on the one hand, together with the fusion method, provides a flexible and economical way to create various multiphoton graph states. Slight modifications of our experimental setup will readily allow the creation of many other graph states, for example, sixqubit linear and Yshaped graph states (see the Supplementary Information). Such a fascinating capacity creates a useful multiqubit graphstate testbed. On the other hand, this feature does not prohibit subsequent applications such as tests of quantum nonlocality^{5,6,7} and inprinciple verifications of linear optical quantuminformation processing tasks where photons need to be eventually detected. Finally, concerning the scalability issue, we refer to ref. 11, which has shown that if combined with quantum memory, the postselection method can even be used for scalable generation of treegraph states using realistic linear optics. Along this line, however, technically extensive efforts still need to be undertaken to make a quantum memory usable for this purpose.
In summary, we have realized two special graph states, the sixphoton GHZ state, the largest photonic Schrödinger cat so far, and the sixphoton cluster state—a stateoftheart oneway quantum computer. We have demonstrated the ability to entangle six photons and to engineer multiqubit graph states, and have created a versatile testbed for experimental investigations of oneway quantum computation^{4}, quantum error correction^{3}, studies of multiparticle entanglement^{1} and foundational tests of quantum physics^{5,6,7,8}. Combined with quantum memory, our experimental method could lead to the generation of largescale treegraph states^{11}. The high efficiency and flexibility of the sixphoton graphstate generation we demonstrated here suggest that photons manipulated with linear optics are promising candidates for engineering of multiqubit graph states. Various applications of our sixphoton graphstate testbed can be imagined. For instance, the sixqubit cluster states allow full implementations of the quantum game of prisoners’ dilemma^{12} and a proofofprinciple demonstration of the basic elements of losstolerant oneway quantum computation^{14}. Most remarkably, the sixqubit starring graph state corresponds to the codeword and encoding procedure of the fivequbit quantum errorcorrection code that is able to correct all onequbit errors^{3}. In addition, our sixphoton cluster state also enables a novel test of nonlocality, namely a GHZ argument of nonlocality for mixed states^{6}. Lastly, the graphstate testbed is well suited for studies of the stabilities of different types of multiparticle entanglement (for example, GHZ and cluster) under the influence of decoherence, which may provide experimental evidence for the surprising conclusion in ref. 8 that genuine entanglement of a macroscopic number of particles is possible and can persist for timescales that are independent of the size of the system.
Methods
Optimizing the sixphoton setup
To get highbrightness entangled photons, similarly to ref. 24 we used the highpower Verdi18/Mira900F femtosecond laser system that outputs a stable pulsed infrared laser with a power of 2.4 W. In addition, we adopted a new threeBBOcrystal configuration for a more flexible optimization of three independent photon pairs. Moreover, the use of narrowband filters in our experiment with peak transmission rates as high as 98% considerably improved the collection efficiency. To make our setup stable, we first made efforts to minimize the fluctuation of the pump power. As shown in Fig. 2, we mounted the LiB_{3}O_{5} (LBO) crystal on a motorized translation stage controlled by a computer program to avoid damage to the crystal by the focusing laser beam. By doing this, we achieved both good upconversion efficiency (∼38%) and long life time for LBO, making it a very stable ultraviolet laser source with a power of 0.91 W that typically drops by less than 1.5% over weeks. Our feedback control system has greatly improved the stability of the downconversion rates: the measured coincidence counts typically fluctuate by ∼3% over two weeks, whereas without it the counts drop significantly after about 15 min (see the Supplementary Information for a curve of the measured twofold coincidences over time). Second, we used a programmable multichannel coincidence unit that allowed us to simultaneously register any possible coincidence detection between the inputs. Aside from the desired sixfold coincidence events, we also registered other combinations (such as single counts and twofold coincidences), so we were able to track the performance of our system in real time. Any power drift could be deduced from the recorded count rates, and finally the sixfold coincidences were corrected for their difference. Thus, we were able to achieve a stable setup, good collection efficiencies and three EPR pairs with high visibilities.
Clusterstate witness construction
The witness for the cluster state (2) can be constructed as follows. Using the results of ref. 26, the observable W_{C}=I/2−C_{6}〉〈C_{6} is a witness detecting genuine multipartite entanglement around the cluster state. Then, we consider the observable
where g_{i} denotes the stabilizing operators of the cluster state (see Fig. 4a). Furthermore, we use A_{0}=I−H H H〉〈H H H−V V V 〉〈V V V , A_{1}=V V V 〉〈V V V −H H H〉〈H H H and , where M_{(i)} is defined as for the GHZ state (see equation (3)). Finally, denotes a cluster state with different signs, namely .
It is clear that , which implies that is a valid witness^{26}. Furthermore, this implies that the fidelity of the cluster state can be estimated as .
The witness (4) detects genuine entanglement from the states of the form ρ(p)=pC_{6}〉〈C_{6}+(1−p)I/64 for p>0.5. The determination of the expectation value of the witness requires six measurement settings, namely , , and . The results are shown in Fig. 4 in the main text and in Supplementary Information, Fig. S3.
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Acknowledgements
We thank H. J. Briegel, D. Browne, L.M. Duan, T. Rudolph and S. Yu for helpful discussions. This work was supported by the National Natural Science Foundation of China, the Chinese Academy of Sciences. This work was also supported by the Alexander von Humboldt Foundation, the Marie Curie Excellence Grant of the EU, the FWF, the DFG and EU (Scala, Olaqui, Prosecco, QICS, Quprodis).
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Lu, C., Zhou, X., Gühne, O. et al. Experimental entanglement of six photons in graph states. Nature Phys 3, 91–95 (2007). https://doi.org/10.1038/nphys507
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