Abstract
Quantum entanglement^{1,2} plays a vital role in many quantuminformation and communication tasks^{3}. Entangled states of higherdimensional systems are of great interest owing to the extended possibilities they provide. For example, they enable the realization of new types of quantum information scheme that can offer higherinformationdensity coding and greater resilience to errors than can be achieved with entangled twodimensional systems (see ref. 4 and references therein). Closing the detection loophole in Bell test experiments is also more experimentally feasible when higherdimensional entangled systems are used^{5}. We have measured previously untested correlations between two photons to experimentally demonstrate highdimensional entangled states. We obtain violations of Belltype inequalities generalized to ddimensional systems^{6} up to d=12. Furthermore, the violations are strong enough to indicate genuine 11dimensional entanglement. Our experiments use photons entangled in orbital angular momentum^{7}, generated through spontaneous parametric downconversion^{8,9}, and manipulated using computercontrolled holograms.
Main
Quantuminformation tasks requiring highdimensional bipartite entanglement include teleportation using qudits^{10,11}, generalized dense coding (that is, with pairs of entangled dlevel systems; ref. 12) and some quantum key distribution protocols^{13}. More generally, schemes such as quantum secret sharing^{14} and measurementbased quantum computation^{15} apply multiparticle entanglement. These are promising applications, especially in view of recent progress in the development of quantum repeaters (see ref. 16 and references therein). However, practical applications of such protocols are only conceivable when it is possible to experimentally prepare, and moreover detect, highdimensional entangled states. Therefore, the ability to verify highdimensional entanglement between physical qudits is of crucial importance. Indeed, much progress has generally been made on the generation and detection of highdimensional entangled states (please see ref. 17 and references within).
Here we report the experimental investigation of highdimensional, twophoton entangled states. We focus on photon orbital angular momentum (OAM) entangled states generated by spontaneous parametric downconversion (SPDC), and demonstrate genuine highdimensional entanglement using violations of generalized Belltype inequalities^{6}. Previously, qutrit Belltype tests have been carried out using photon OAM to verify threedimensional entanglement (see ref. 18 and references within). In addition to testing whether correlations in nature can be explained by local realist theories^{19}, the violation of Belltype inequalities may be used to demonstrate the presence of entanglement. Belltype experiments have been carried out using twodimensional subspaces of the OAM state space of photons^{20,21} and experimentalists have demonstrated twodimensional entanglement using up to 20 different twodimensional subspaces^{22}. Careful studies have also been carried out to describe how specific detector characteristics bound the dimensionality of the measured OAM states in photons generated by SPDC using Shannon dimensionality^{23}.
Our experimental study of highdimensional entanglement is based on the theoretical work of Collins et al. ^{6}, which was applied in experiments for qutrits encoded in the OAM states of photons^{18,24}. We encode qudits using the OAM states of photons, with eigenstates defined by the azimuthal index ℓ. These states arise from the solution of the paraxial wave equation in its cylindrical coordinate representation, and are the Laguerre–Gaussian modes L G_{p,ℓ}, so called because they are light beams with a Laguerre–Gaussian amplitude distribution.
In our setup (Fig. 1), OAM entangled photons are generated through a frequencydegenerate typeI SPDC process, and the OAM state is manipulated with computercontrolled spatial light modulators (SLMs) acting as reconfigurable holograms. Conservation of angular momentum ensures that, if the signal photon is in the mode specified by ℓ〉, the corresponding idler photon can only be in the mode −ℓ〉. Assuming that angular momentum is conserved^{9}, a pure state of the twophoton field produced will have the form
where subscripts A and B label the signal and idler photons respectively, c_{ℓ}^{2} is the probability to create a photon pair with OAM ±ℓℏ and ℓ〉 is the OAM eigenmode with mode number ℓ.
It has been shown^{6} that, for correlations that can be described by theories based on local realism^{1}, a family of Belltype parameters S_{d} satisfies the inequalities
Alternatively, if quantum mechanics is assumed to hold, then the violation of an inequality of type (1) indicates the presence of entanglement. S_{d} can be expressed as the expectation value of a quantum mechanical observable, which we denote as . The expressions for S_{d}, and the operators and are provided in Supplementary Section SI.
The parameters S_{d} are calculated using coincidence probabilities for measurements made locally by two observers, Alice and Bob, on their respective subsystems, which in our case are the signal and idler photons from the SPDC source. Alice’s detector has two settings labelled by a∈{0,1} with d outcomes for each setting, and similarly for Bob’s detector with settings b∈{0,1}. The measurement bases corresponding to the detector settings of Alice and Bob are defined as
where v and w both run from 0 to d−1 and denote the outcomes of Alice’s and Bob’s measurements respectively, and the parameters α_{0}=0, α_{1}=1/2, β_{0}=1/4 and β_{1}=−1/4.
The measurement bases {v〉_{a}^{A}} and {w〉_{a}^{A}} have been shown^{5,25} to maximize the violations of inequality (1) for the maximally entangled state of two ddimensional systems given by It turns out that we are able to parameterize these ddimensional measurement basis states with ‘mode analyser’ angles θ_{A} and θ_{B}, and write them in the form
where
The function g(ℓ) is defined as
where [x] is the integer part of x, and u(ℓ) is the discrete unit step function.
Figure 2 shows an example of the experimental data points for the selfnormalized coincidence rates as function of the relative angle (θ_{A}−θ_{B}) using d=11 (see also Supplementary Fig. S1). For a maximally entangled state
where h(ℓ)=1 for all ℓ when d is odd, and h(ℓ≠0)=1, h(0)=1 when d is even, the coincidence rate of detecting one photon in state θ_{A}〉 and the other in state θ_{B}〉 is proportional to
The key result of our paper is shown in Fig. 3, which shows a plot of experimental values of parameter S_{d} as a function of the number of dimensions d. The plot compares theoretically predicted violations for a maximally entangled state, the experimental readings and the local hidden variable (LHV) limit. The maximum possible violations (shown in Supplementary Table S1) are slightly larger than the corresponding violations produced by a maximally entangled state. Violations persist up to as much as d=12 when entanglement concentration^{26} is applied. We find S_{11}=2.39±0.07 and S_{12}=2.24±0.08, which clearly violate S_{d}≤2 (see also Supplementary Table S4). In the corresponding experiment using L G_{p,ℓ} modes with only p=0, violations are obtained up to d=11. Without entanglement concentration, we observe violations only up to d=9 (please see Supplementary Fig. S2 in Section SIV). Above d∼11, the strength of the signal becomes so low that noise begins to overshadow the quantum correlations. In Fig. 2, the theoretical prediction in equation (5) for a state with maximal 11dimensional entanglement is fitted to the experimental coincidence data obtained using the mode analyser settings defined in equation (4) for d=11, with only the vertical offset and amplitude left as free parameters. The observed fringes are seen to closely match those theoretically obtained for a state with maximal 11dimensional entanglement.
The violation of a Bell inequality in d×d dimensions directly indicates that the measured state was entangled. It remains to determine how many dimensions were involved in the entanglement. Measuring the coincidence probabilities, that is, of there being the joint state (Fig. 4), together with the parameters S_{d} for different d, can be seen as a partial tomography of the SPDC source state. Numerical investigations indicate that a state with the experimentally observed coincidence probabilities and parameters S_{2},S_{3},…,S_{11} must contain genuine 11dimensional entanglement. In other words, it is not possible to obtain the observed levels of violation with a state that contains entanglement involving only 10 dimensions or less. Our analysis assumes a special form of the states, based on the coincidence measurement results shown in Fig. 4. Further details are given in Supplementary Section SII.
Our results hold much promise for applications requiring entangled qudits in general. As mentioned earlier, progress in the development of quantum repeaters (see ref. 16 and references therein) would make quantum key distribution using highdimensional entangled states^{13} a possible application. Conventional quantum communication will fail for sufficiently large transmission distances because of loss, and quantum repeaters are one possible solution to this problem. Although experimental quantum key distribution has been demonstrated with OAM qutrits^{24}, our findings provide experimental evidence that such protocols could be implemented using photons entangled in OAM in up to 11 dimensions, resulting in a considerable increase in information coding density.
A possible extension to our work would be to investigate the generation of multiphoton, highdimensional OAM entanglement. We can conceive of achieving this using a cascade of downconversion crystals for generating multipartite entangled photons, which has been done for polarization entangled photons^{17}. It also seems to be within reach to combine the highdimensional photon OAM entanglement with entanglement in the polarization and path degrees of freedom, creating even larger hyperentangled states (see ref. 17 and references within).
On a more fundamental note, Bell test experiments carried out so far have one or both of two main loopholes, namely the locality and detection loopholes. However, a recent theoretical work reveals that even lowdimensional qudits can provide a significant advantage over qubits for closing the detection loophole^{5}. In fact, it was found that as much as 38.2% loss can be tolerated using fourdimensional entanglement. Our results raise interesting possibilities regarding the role higherdimensional entangled qudits could play in closing this loophole. We emphasize that neither the detection nor the locality loophole has been closed in our experiments, because the overall efficiency of our experimental setup is 1–2%, and the switching time for our measurement devices (SLMs) is of the order of tens of milliseconds. However, closing these loopholes was not the immediate goal of our experiments. We are instead using the violation of Bell inequalities, up to fair sampling assumptions, as a means of verifying the presence of highdimensional entanglement, within the framework of quantum mechanics.
In summary, we have been able to experimentally demonstrate violations of Belltype inequalities generalized to ddimensional systems^{6} with up to d=12, enough to indicate genuine 11dimensional entanglement in the OAM of signal and idler photons in parametric downconversion. It seems that this could be extended to even higher dimensions by using a brighter source of entangled photons.
Methods
In our experiments, we use computercontrolled SLMs (Hamamatsu) operating in reflection mode with a resolution of 600×600 pixels. In the detection, the SLMs are prepared in the states defined in equation (4) respectively. An SLM prepared in a given state transforms a photon in that state to the Gaussian ℓ=0〉 mode. The reflected photon is then coupled into a singlemode fibre which feeds a single photon detector. As only the ℓ=0〉 mode couples into the fibre, a count in the detector indicates a detection of the state in which the SLM was prepared. The hologram generation algorithm introduced in ref. 27 is applied to configure the SLMs.
Figure 1 shows a schematic diagram of the experimental setup as well as examples of SLM settings used where d=1. For the SPDC, we use a pump beam, with ℓ=0, produced by a frequencytripled, modelocked NdYAG laser with an average output power of 150 mW at 355 nm. The collimated laser beam is normally incident on a 3mmlong βbarium borate (BBO) crystal cut for typeI collinear phase matching. A 50:50 beam splitter (BS) then separates the copropagating OAM entangled photons probabilistically into the signal and idler paths. Spectral filters with 10 nm bandwidth are used to reduce the detection of noise photons. The coincidence resolving time is 10 ns and an integration time of 20 s is used for the measurements.
For tests within a ddimensional subspace and for odd d, we choose the modes ℓ=−(d−1)/2,…,0,…,(d−1)/2 as the computational basis states j〉 in equations (2) and (3), where j=0,…,d−1. For even d, we use ℓ=−d/2,…,−1,1,…,d/2, omitting the ℓ=0 mode. A projection of the SPDC output state onto a ddimensional subspace results in a nonmaximally entangled state owing to the limited spiral bandwidth^{28}. To enhance the entanglement, we use the socalled Procrustean method of entanglement concentration^{26}. This is generally done by means of a filtering technique that equalizes the mode amplitudes, thereby probabilistically enhancing the entanglement of the twophoton state^{29}. This can be achieved by applying local operations to one or both of the signal and idler photons. We choose local operations matched to the spiral bandwidth measurement for our SPDC source (please see Supplementary Section SIV), so as to obtain a close approximation to a maximally entangled state. The method applied in ref. 30 uses lenses for equalizing amplitudes in a superposition of three OAM modes. We however use alterations of the diffraction efficiencies of blazed phase gratings in the SLMs to achieve this goal for up to 14 modes. Figure 4 contrasts the results of coincidence measurements with and without Procrustean filtering, with the SLMs in the state where ℓ_{A},ℓ_{B}∈{−5,…,+5}. The disadvantage of the Procrustean method is the associated reduction in the number of detected photons (see Supplementary Section SIII for further details).
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Acknowledgements
We acknowledge suggestions from S. M. Barnett. This work was funded by the Engineering and Physical Sciences Research Council (EPSRC). A.C.D. acknowledges funding support from the Scottish Universities Physics Alliance (SUPA).
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A.C.D. and E.A. devised the concept of the experiment. E.A. supervised the theoretical aspects of the project. G.S.B. and M.J.P. advised on aspects of experimental design. A.C.D. and J.L. carried out the experiment. A.C.D. and E.A. wrote the paper with contributions from all authors.
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Dada, A., Leach, J., Buller, G. et al. Experimental highdimensional twophoton entanglement and violations of generalized Bell inequalities. Nature Phys 7, 677–680 (2011). https://doi.org/10.1038/nphys1996
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