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Experimental High-Dimensional Two-Photon Entanglement and Violations of Generalized Bell Inequalities

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Experimental High-Dimensional Two-Photon Entanglement and Violations of Generalized Bell Inequalities LETTERS PUBLISHED ONLINE: 8 MAY 2011 | DOI: 10.1038/NPHYS1996 Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities Adetunmise C. Dada1*, Jonathan Leach2, Gerald S. Buller1, Miles J. Padgett2 and Erika Andersson1 Quantum entanglement1,2 plays a vital role in many by local realist theories19, the violation of Bell-type inequalities may quantum-information and communication tasks3. Entangled be used to demonstrate the presence of entanglement. Bell-type states of higher-dimensional systems are of great interest experiments have been carried out using two-dimensional sub- owing to the extended possibilities they provide. For exam- spaces of the OAM state space of photons20,21 and experimentalists ple, they enable the realization of new types of quantum have demonstrated two-dimensional entanglement using up to 20 information scheme that can offer higher-information-density different two-dimensional subspaces22. Careful studies have also coding and greater resilience to errors than can be achieved been carried out to describe how specific detector characteristics with entangled two-dimensional systems (see ref. 4 and bound the dimensionality of the measured OAM states in photons references therein). Closing the detection loophole in Bell generated by SPDC using Shannon dimensionality23. test experiments is also more experimentally feasible when Our experimental study of high-dimensional entanglement is higher-dimensional entangled systems are used5. We have based on the theoretical work of Collins et al.6, which was applied in measured previously untested correlations between two experiments for qutrits encoded in the OAM states of photons18,24. photons to experimentally demonstrate high-dimensional We encode qudits using the OAM states of photons, with eigenstates entangled states. We obtain violations of Bell-type inequalities defined by the azimuthal index `. These states arise from the generalized to d-dimensional systems6 up to d D 12. Further- solution of the paraxial wave equation in its cylindrical co-ordinate more, the violations are strong enough to indicate genuine representation, and are the Laguerre–Gaussian modes LGp;`, so 11-dimensional entanglement. Our experiments use photons called because they are light beams with a Laguerre–Gaussian entangled in orbital angular momentum7, generated through amplitude distribution. spontaneous parametric down-conversion8,9, and manipulated In our set-up (Fig. 1), OAM entangled photons are generated using computer-controlled holograms. through a frequency-degenerate type-I SPDC process, and the Quantum-information tasks requiring high-dimensional bipar- OAM state is manipulated with computer-controlled spatial tite entanglement include teleportation using qudits10,11, general- light modulators (SLMs) acting as reconfigurable holograms. ized dense coding (that is, with pairs of entangled d-level sys- Conservation of angular momentum ensures that, if the signal tems; ref. 12) and some quantum key distribution protocols13. photon is in the mode specified by j`i, the corresponding idler More generally, schemes such as quantum secret sharing14 and photon can only be in the mode j−`i. Assuming that angular measurement-based quantum computation15 apply multiparticle momentum is conserved9, a pure state of the two-photon field entanglement. These are promising applications, especially in view produced will have the form of recent progress in the development of quantum repeaters (see `D1 ref. 16 and references therein). However, practical applications X j9i D c`j`iA ⊗|−`iB of such protocols are only conceivable when it is possible to `=−∞ experimentally prepare, and moreover detect, high-dimensional entangled states. Therefore, the ability to verify high-dimensional where subscripts A and B label the signal and idler photons 2 entanglement between physical qudits is of crucial importance. respectively, jc`j is the probability to create a photon pair with Indeed, much progress has generally been made on the generation OAM `hN and j`i is the OAM eigenmode with mode number `. and detection of high-dimensional entangled states (please see It has been shown6 that, for correlations that can be described by 1 ref. 17 and references within). theories based on local realism , a family of Bell-type parameters Sd Here we report the experimental investigation of high- satisfies the inequalities dimensional, two-photon entangled states. We focus on photon S(local realism) ≤ 2; for all d ≥ 2 (1) orbital angular momentum (OAM) entangled states generated by d spontaneous parametric down-conversion (SPDC), and demon- Alternatively, if quantum mechanics is assumed to hold, then the strate genuine high-dimensional entanglement using violations violation of an inequality of type (1) indicates the presence of 6 of generalized Bell-type inequalities . Previously, qutrit Bell-type entanglement. Sd can be expressed as the expectation value of tests have been carried out using photon OAM to verify three- a quantum mechanical observable, which we denote as Sˆd . The dimensional entanglement (see ref. 18 and references within). In expressions for Sd , Sˆd and the operators Sˆ2 and Sˆ3 are provided in addition to testing whether correlations in nature can be explained Supplementary Section SI. 1SUPA, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK, 2Department of Physics and Astronomy, SUPA, University of Glasgow, Glasgow G12 8QQ, UK. *e-mail: [email protected]. NATURE PHYSICS j VOL 7 j SEPTEMBER 2011 j www.nature.com/naturephysics 677 © 2011 Macmillan Publishers Limited. All rights reserved. LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS1996 Examples of hologram states for d = 11: Desired intensity Phase SLM A f = 600 mm f = 3.2 mm Ultraviolet pump, += 355 nm θ a = 0 π 〉 ⎪ 〉 a = 0 ⎪ A = 2 /11 (or v = 1 A ) Coincidence f = 300 mm Spectral Single C (θθ a , b) 50:50 A B filters mode fibres BBO, type 1 BS nonlinear crystal Desired intensity Phase += f = 600 mm f = 3.2 mm SLM B 0 2π ⎪θ b = 0 π 〉 ⎪ 〉b = 0 B = 3 /2 (or w = 3 B ) D ; D θ a;θ b Figure 1 j Schematic representation of experimental set-up for violations of Bell-type inequalities. C(Aa v Bb w) or C( A B ) is the coincidence count j iA jθ ai j iB jθ bi rate when SLM A is in state v a or A and SLM B is in state w b or B respectively. The parameters Sd are calculated using coincidence probabilities Figure 2 shows an example of the experimental data points for for measurements made locally by two observers, Alice and Bob, the self-normalized coincidence rates as function of the relative on their respective subsystems, which in our case are the signal angle (θA −θB) using d D 11 (see also Supplementary Fig. S1). For a and idler photons from the SPDC source. Alice's detector has maximally entangled state two settings labelled by a 2 f0; 1g with d outcomes for each Td=2U setting, and similarly for Bob's detector with settings b 2 f0;1g. The 1 X j8i D p h(`)j`i ⊗|−`i measurement bases corresponding to the detector settings of Alice A B d `=−[d=2U and Bob are defined as ` D ` ` 6D D D − where h( ) 1 for all when d is odd, and h( 0) 1, h(0) 1 1 Xd 1 2π jviA D p exp i j(v Cα ) jji (2) when d is even, the coincidence rate of detecting one photon in state a a jθ i jθ i d jD0 d A and the other in state B is proportional to cos(d(θA −θB))−1 d−1 π C(θ ;θ ) D jhθ jhθ jj8ij2 / (5) 1 X 2 A B A B 3 j iB D p − Cβ j i d Tcos(θA −θB)−1U w b exp i j( w b) j (3) d jD0 d The key result of our paper is shown in Fig. 3, which shows a where v and w both run from 0 to d −1 and denote the outcomes plot of experimental values of parameter Sd as a function of the of Alice's and Bob's measurements respectively, and the parameters number of dimensions d. The plot compares theoretically predicted α0 D 0, α1 D 1=2, β0 D 1=4 and β1 D −1=4. violations for a maximally entangled state, the experimental fj iAg fj iAg 5,25 The measurement bases v a and w a have been shown readings and the local hidden variable (LHV) limit. The maximum to maximize the violations of inequality (1) for the maximally possible violations (shown in Supplementary Table S1) are slightly entangledp state of two d-dimensional systems given by j i D larger than the corresponding violations produced by a maximally = Pd−1 j i ⊗j i : D (1 d) jD0 j A j B It turns out that we are able to parameterize entangled state. Violations persist up to as much as d 12 when 26 these d-dimensional measurement basis states with `mode analyser' entanglement concentration is applied. We find S11 D 2:390:07 angles θA and θB, and write them in the form `DCTd=2U 1.0 1 X d = 11 scan jviA ≡ jθ ai D p expiθ ag(`)j`i; and a A A Fitted curve d `=−[d=2U 0.8 `DCT d U 0.6 1 X2 jwiB ≡ jθ bi D p expiθ bg(`)j`i (4) b B B 0.4 d `=−[ d U 2 where Normalized coincidences 0.2 θ a D C = π= A (v a 2)2 d 0 θ b D [− C = − bU π= ¬π ¬π/2 0 π/2 π B w 1 4( 1) 2 d θθ Relative orientation of analysers ( A ¬ B) The function g(`) is defined as d Figure 2 j Coincidence count rate (self-normalized) as a function of the g(`) D `C C(d mod 2)u(`) θ −θ 2 relative orientation angle between state analysers ( A B).
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