Abstract
Coherent manipulation of a large number of qubits and the generation of entangled states between them has been an important goal and benchmark in quantum information science, leading to various applications such as measurementbased quantum computing^{1} and highprecision quantum metrology^{2}. However, the experimental preparation of multiparticle entanglement remains challenging. Using atoms^{3,4}, entangled states of up to eight qubits have been created, and up to six photons^{5} have been entangled. Here, by exploiting both the photons’ polarization and momentum degrees of freedom, we experimentally generate hyperentangled six, eight and tenqubit Schrödinger cat states with verified genuine multiqubit entanglement. We also demonstrate superresolving phase measurements enhanced by entanglement, with a precision to beat the standard quantum limit. Modifications of the experimental setup would enable the generation of other graph states up to ten qubits. Our method offers a way of expanding the effective Hilbert space and should provide a versatile testbed for various quantum applications.
Main
Control of single photonic qubits using linear optics has been an appealing approach to implementing quantum computing^{6}. Experiments in recent years have demonstrated the photons’ extremely long decoherence time^{7}, fast clock speed^{8}, a series of controlled quantum logic gates^{9,10} and algorithms^{8,11,12} and the generation of multiqubit entangled states^{5}. A significant challenge, however, lies in making experimentally accessible sources of photonic multiqubit states. This is because, on the one hand, the probabilistic nature of spontaneous parametric downconversion^{13} represents a bottleneck with regard to the attainable brightness and fidelity of multiphoton states based on it: manipulating seven photons or more seems an insurmountable challenge with present technology. On the other hand, triggered singlephoton sources from independent quantum dots or other emitters still suffer from spectral and temporal distinguishability, which prevents upscaling.
There is, however, a way to experimentally control more effectively qubits, by exploiting hyperentanglement^{14}—the simultaneous entanglement in the multiple degrees of freedom that naturally exist for various physical systems. For instance, one can encode quantum information not only in the polarization of a single photon, but also in its spatial modes^{15}, arrival time^{15} or orbital angular momentum^{16}. Recently, hyperentangled photonic states^{15} have been experimentally realized, and shown to offer significant advantages in quantum superdense coding^{17,18}, enhanced violation of local realism^{19}, efficient construction of cluster states^{20,21} and multiqubit logic gates^{10}.
Although the largest hyperentangled state^{15} realized so far has expanded the Hilbert space up to 144 dimensions, it is a product state of twoparty entangled states and does not involve multipartite entanglement. Other schemes^{20,21} for creating hyperentanglement have been limited by the technical problem of photonic subwavelength phase stability and seem infeasible to generate larger states than the twophoton fourqubit ones. Here, we will describe a method that overcomes these limitations, and the experimental generation of hyperentangled six, eight and tenqubit photonic Schrödinger cat states.
The Schrödinger cat states, also technically known as Greenberger–Horne–Zeilinger states^{22}, involve an equal superposition of two maximally different quantum states. They are of particular interest in quantum mechanics and find wide applications in quantum information processing. Our experiment aims to create the cat state in the form:
where H and V denote horizontal and vertical polarization and H′ and V ′ label two orthogonal spatial modes (or momentums) of the photons. The state (1) shows maximal entanglement between all photons’ polarization and spatial qubits.
Our first experimental step is to generate polarizationentangled nqubit cat states . Two pairs of entangled photons are produced by spontaneous parametric downconversion^{13} in the state in paths 2–3 and 4–5, and a pseudosingle photon source^{23} is prepared in the polarization state in path 1 (see Fig. 1a and the Methods section). Photons 3 and 4 are superposed on a polarizing beam splitter (PBS_{1}), and then are further combined with photon 1 on PBS_{2}. Fine adjustments of the delay between the different paths are made to ensure that the photons arrive at the PBSs simultaneously. Furthermore, the photons are spectrally filtered and detected by singlemode fibrecoupled singlephoton detectors for good spatial and temporal overlap. As the PBSs transmit H and reflect V polarization, it can be concluded that a coincidence detection of the five output photons implies that all of the photons are either H or V polarized—the two cases are quantum mechanically indistinguishable—therefore projecting them in the cat state . It is easy to check that, in a similar way, if we combine only photons 1 and 4 (3 and 4) on PBS_{2} (PBS_{1}), entangled cat states between the three photons 1–4–5 (the four photons 2–3–4–5) can be created by postselection.
Next, we grow the polarizationencoded nqubit states into doublesized 2nqubit cat states by planting spatial modes on them. This again exploits the PBSs. Consider a polarized singlephoton qubit in the state αH〉+βV 〉 that passes through a PBS (see Fig. 1b). The PBS separates the photon into two possible spatial modes H′ and V ′, according to their polarization H and V respectively; indeed, this forms the basis of the PBS as an instrument for measuring polarization. The state of this single photon can now be written as αH〉H′〉+βV 〉V ′〉, an entangled state between its polarization and spatial degree of freedom. It is straightforward to extend this method to the nphoton state Cat〉_{p}^{n}; thereby, the hyperentangled 2nqubit cat state (1) can be created. Besides the cat states, we note that this method can also be flexibly modified for the generation of other graph states^{24}, which are central resources in measurementbased quantum computing (see Supplementary Fig. S1).
With multiple degrees of freedom carrying the quantum information in a single photon, measurements of the composite quantum states now become a bit trickier, as it is necessary to read out one degree of freedom without disturbing another one. Figure 1b illustrates the apparatus for simultaneously yet independently measuring both the polarization and spatial qubits on the basis of 0〉/1〉 and (0〉±e^{iθ}1〉) (here we denote H〉=H′〉 as logic 0〉 and V 〉=V ′〉 as 1〉). Specifically, the measurement of the spatial qubit uses an optical interferometer combing the two paths onto a nonpolarizing beam splitter (NBS) with an adjustable delay between these two paths that controls the relative phase θ. After this interferometer, the polarization information is then read out by using a combination of a quarterwave plate, a halfwave plate and a PBS in front of the singlephoton detectors, projecting the polarization states into (H〉±e^{iθ}V 〉). Experimentally, however, it is difficult to directly implement the interferometer in Fig. 1b, because it is sensitive to path length instability of the order of the photon’s wavelength. To overcome this problem, we construct intrinsically stable Sagnaclike interferometers with beamsplitter cubes that are half PBScoated and half NBScoated (see Fig. 1c). The longterm stabilities and the high visibilities of the five interferometers constructed in our experiment are shown in Supplementary Fig. S2.
As a stepbystep approach, we begin with the creation of the hyperentangled sixqubit cat state Cat〉^{6} and eightqubit cat state Cat〉^{8}. To analyse the experimentally produced states, we first look at the measurement results in the 0〉/1〉 basis, as shown in Fig. 2a,b for Cat〉^{6} and Cat〉^{8}, respectively. For ideal cat states, the desired combinations in this basis should in principle be only and . This is confirmed by the experimental data in Fig. 2a,b, showing that these two terms dominate the overall coincidence events, with a signaltonoise ratio (defined as the ratio of the average of the desired components to that of the other nondesired ones) of 85:1 to 1,100:1 for the states Cat〉^{6} and Cat〉^{8}, respectively. We note that the undesired noise, noticeably located in the diagonal line of Fig. 2a,b, mainly arises from the doublepair emission of entangled photons.
Although the above data determine the population in the 0〉/1〉 basis of the cat states, they are not sufficient to reveal their coherence properties. We took therefore additional measurements in the basis and . In this new basis, the cat state Cat〉^{n} can be written in the form of ; thus, the probabilities of creating the components , and hence the experimentally observed coincidence events should vary as ∼(1±cosn θ). From these measurements, one can determine the expectation values of the spin observable: , where M_{θ}=cosθ σ_{x}+sinθ σ_{y}, which oscillates n times sinusoidally over a single cycle of 2π. Indeed, this can arise only from coherent superposition between the and component of the cat state and serves as a characteristic signature of nqubit coherence^{3}.
Figure 2c,d shows the experimentally obtained expectation values as a function of θ (0≤θ≤2π) and the fitted sinusoidal fringes. The fringes clearly show the n θ oscillation, with a visibility of 0.527±0.002 and 0.67±0.01 for the six and eightqubit cat states, respectively, confirming the coherence between all effective n qubits encoded with either polarization or spatial information. We note that the reduction of the visibilities is caused by, besides the abovementioned doublepair photon emission, also the imperfections of photon overlapping at the PBSs and NBSs.
From the data shown in Fig. 2, we can further determine the fidelities of the cat states and detect the presence of genuine multipartite entanglement^{25}. The fidelity—a measure of the extent to which the desired state is created—is the overlap of the experimentally produced state with the ideal one: F(ψ〉)=〈ψρ_{exp}ψ〉. For the cat state, ψ〉〈ψ can be decomposed as^{5} , corresponding to measurements in the basis of 0〉/1〉 and (0〉±e^{i(kπ/n)}1〉). Figure 2a–d shows the experimental results, from which the fidelities of the six and eightqubit cat states can be determined: F(Cat〉^{6})=0.6308±0.0015, F(Cat〉^{8})=0.776±0.006. The notion of genuine multipartite entanglement characterizes whether generation of the state requires interaction of all parties, distinguishing the experimentally produced state from any incompletely entangled state. For cat states, it is sufficient for the presence of genuine multipartite entanglement that their fidelities exceed the threshold of 0.5 (ref. 25). Thus, with high statistical significance, genuine nqubit entanglement of the cat states created in our experiment is confirmed. We note that the fidelity of the sixqubit cat state is considerably lower than that of the eightqubit state, which is due to the fact that the generation of the former involves a faint coherent laser light, which introduces more noise than the configuration of the latter. It is worth mentioning here an advantage that the hyperentanglement brings—our new sixqubit cat state not only has a higher fidelity than the previous sixphoton cat state^{5}, but also its count rate reaches ∼200 s^{−1}, some 4 orders of magnitude brighter than the sixphoton coincidence.
The fringes in Fig. 2c,d show n θ phase dependencies, which are n times more sensitive to phase changes than that of a single qubit, highlighting the potential use of the cat states for superresolving phase measurements^{2}. Suppose we need to estimate the phase ϕ of a process 0〉→0〉 and 1〉→e^{iϕ}1〉. A simple initial probe state would evolve into and the phase can be estimated from the overlap 〈ψ_{i}ψ_{f}〉^{2}=[1+cos(ϕ)]/2. Provided with N probe states, the uncertainty of the phase estimation is —the standard quantum limit. Applying the same dynamics on N qubits in a cat state , it will evolve to . Then, measuring the overlap or the mean value leads to dependence on cos(N ϕ), from which the phase can be determined with an error of Δϕ=1/N (refs 2, 3, 26). The 1/N scaling of the precision is called the Heisenberg limit, as it coincides with the limit imposed by the uncertainty principle. As shown in Fig. 2c,d, we have observed the expected ntimes phase superresolution for the six and eightqubit cat states, with a visibility of 0.527±0.002 and 0.67±0.01, respectively. These values are larger than the threshold of for phase superresolution, and taking into account the catstate preparation efficiency of 1/2 caused by the PBS filtering (the efficiency can be improved to unity in principle^{6}), we find that the eightqubit cat state can achieve a phase sensitivity 1.13±0.02 times greater than the standard quantum limit (see Supplementary Information). This represents entanglementenhanced quantum metrology with the largest number of qubits so far. However, as with all photonic experiments^{27}, the singlephoton detection efficiency needs to be improved significantly for a practical implementation of Heisenberglimited measurements.
Now we proceed with describing the generation and analysis of the tenqubit cat state Cat〉^{10}, which uses the full setup shown in Fig. 1. As a result of the probabilistic nature of spontaneous parametric downconversion, the coincidence count rate of the tenqubit state is as low as 0.021 Hz, 1/160 (1/11,000) of that of the eightqubit (sixqubit) state. Measurement results along the 0〉/1〉 basis are shown in Fig. 3a with all 2^{10}=1,024 possible combinations plotted, giving a signal to noise ratio (defined as above) of 940:1. We further take measurements in the (0〉±e^{iθ}1〉) basis, where θ is chosen as: θ=kπ/10, k=0,1,…,9. The measured expectation values of the observable are listed in Fig. 3b with an average absolute value of 0.475—this can also be seen equivalently as the fringe visibility shown in Fig. 2c,d. We can thus calculate the state fidelity: F(Cat〉^{10})=0.561±0.019, which is above the threshold of 0.5 by more than three standard deviations, thus establishing the presence of genuine tenqubit entanglement after postselection. These data are further analysed using an optimized entanglement witness method (see Supplementary Information), which, with even higher significance, confirms that the entanglement truly involves all ten qubits.
We have experimentally analysed the hyperentangled six, eight and tenqubit photonic Schrödinger cat states and their phase superresolution. These results represent the largest entangled state realized so far, expanding the effective Hilbert space up to 1,024 dimensions. The cat states demonstrated here, together with other graph states technically feasible within our experimental method (see Supplementary Fig. S1), create a versatile testing ground for the study of nonlocality^{28}, multipartite entanglement and many quantum information protocols. Indeed, by taking advantage of the hyperentanglement, it is now possible to reach some experimental regimes that were hardly accessible before, for instance, demonstrations of the robustness of anyonic braiding^{29} and topological clusterstate encoding^{30} that require manipulation of 7–10 qubits. It will be interesting in future work to further exploit quantum particles’ other degrees of freedom, such as arrival time and orbital angular momentum, to create larger multidimensional entangled states, aiming for more efficient quantum information processing.
Methods
The photon pairs and single photons for the later observation of the cat state are generated as follows: as shown in Fig. 1a, a femtosecond infrared laser is attenuated to be a weak coherent photon source that has a very small probability (p∼0.03) of containing a single photon for each pulse, and prepared in the superposition state in path 1. Meanwhile, a pulsed ultraviolet (ultraviolet) laser, which is upconverted from the intense infrared laser, passes through two βbarium borate crystals, generating two pairs of entangled photons in the state in paths 2–3 and 4–5. The photon pairs have an average twophoton coincidence count rate of 2.4×10^{4} s^{−1} and a visibility of 0.92 in the H/V basis and 0.90 in the basis.
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Acknowledgements
We thank J.P. Dowling and S.J. van Enk for helpful discussions. This work was supported by the National Natural Science Foundation of China, the Chinese Academy of Sciences and the National Fundamental Research Program (under Grant No 2006CB921900). This work was also supported by the Alexander von Humboldt Foundation, the ERC, the FWF (START prize) and the EU (SCALA, OLAQUI, QICS).
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C.Y.L., W.B.G. and J.W.P. conceived the research; W.B.G., X.C.Y., P.X., A.G., Y.A.C. and C.Z.P. carried out the experiment; O.G. contributed theoretical analytic tools; W.B.G., O.G. and C.Y.L. analysed the data; C.Y.L., O.G., W.B.G. and J.W.P. wrote the paper; J.W.P. and Z.B.C. supervised the whole project.
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Gao, W., Lu, C., Yao, X. et al. Experimental demonstration of a hyperentangled tenqubit Schrödinger cat state. Nature Phys 6, 331–335 (2010). https://doi.org/10.1038/nphys1603
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