## Abstract

Entanglement of the properties of two separated particles constitutes a fundamental signature of quantum mechanics and is a key resource for quantum information science. We demonstrate strong Einstein, Podolsky, and Rosen correlations between the angular position and orbital angular momentum of two photons created by the nonlinear optical process of spontaneous parametric down-conversion. The discrete nature of orbital angular momentum and the continuous but periodic nature of angular position give rise to a special sort of entanglement between these two variables. The resulting correlations are found to be an order of magnitude stronger than those allowed by the uncertainty principle for independent (nonentangled) particles. Our results suggest that angular position and orbital angular momentum may find important applications in quantum information science.

In 1935, Einstein, Podolsky, and Rosen (EPR) proposed a Gedanken experiment that was intended to show that quantum mechanics is incomplete (*1*). Their proposal supposes the existence of two spatially separated particles that are perfectly correlated in both position and momentum. Measurement of the position (or alternatively the momentum) of one particle would then determine instantaneously the position (or momentum) of the second particle. The ability to infer either the position or the momentum of the second particle from a distant measurement on the first seems to imply that both of these quantities must have been predetermined. However, quantum theory (and specifically the uncertainty principle) does not allow the simultaneous, exact knowledge of two noncommuting observables, such as position and momentum, as seems to be required for the second particle. A demonstration of EPR correlations establishes either that quantum mechanics is incomplete, in that systems possess additional hidden variables, or that quantum mechanics is nonlocal, in that measurement of the position or momentum of either particle results in an instantaneous uncertainty of the momentum or position, respectively, of both (*2*).

In 1964, Bell deduced an inequality that distinguishes the predictions of quantum theory from those of any local hidden variable theory (*3*, *4*). Since that time, many experiments have been performed that have decided strongly in favor of quantum theory (*5*, *6*). These Bell-type tests apply only to discrete state-spaces, originally of two dimensions or, more recently, to three or higher dimensions (*7*–*9*). In contrast, EPR correlations provide a demonstration of entanglement both for discrete and continuous variables, such as energy and time (*10*), position and linear momentum (*11*), spatial modes (*12*, *13*), and images (*14*).

In addition to linear momentum, light may also carry angular momentum. The spin angular momentum is manifest as the polarization of light and is described completely within a two-dimensional Hilbert space. However, light beams can also carry a measurable orbital angular momentum that results from their helical phase structure. This phase structure can be described by *15*, *16*), where ϕ is the azimuthal angle and *ħ* is Planck’s constant *h* divided by 2π. For restricted subspaces of two or three dimensions, the orbital angular momentum variable has previously been shown to be an entangled property of down-converted photon pairs (*17*, *18*) and to violate a Bell-type inequality (*19*, *20*). However, these present measurements are different in that they are performed in conjunction with the measurement of angular position, the variable conjugate to angular momentum, and hence demonstrate the existence of EPR correlations.

The probability distributions for linear momentum and position are simply Fourier transforms of each other, and the relationship between the standard deviations of their distributions is expressed in the familiar Heisenberg uncertainty relation Δ*x*Δ*p* ≥ ÿℏ/2. In contrast, angle is 2π periodic, and so the Fourier relation between angular position and angular momentum has a different form (*21*, *22*). The periodic, and therefore bounded, angular variable is expressed as a discrete and unbounded Fourier series of angular momenta. Indeed, it is this angular periodicity that gives the quantized—that is, discrete—nature of angular momentum.

We established experimentally that entanglement, as manifest by EPR correlations, exists for angular variables (*23*). Our apparatus is based on parametric down-conversion, in which a quasi-continuous-wave, mode-locked ultraviolet pump beam at 355 nm is incident on a 3-mm length of nonlinear crystal [β-barium borate (BBO)] (Fig. 1) (*24*).

Key to our approach is that just as the spatial light modulator (SLM) can be programmed to transform the fundamental near-Gaussian mode emitted from a fiber into any complex spatial mode of choice, if operated in reverse the same hologram can efficiently couple the same complicated mode distribution back into a fiber. In this configuration, the SLM acts as a mode filter, allowing the transverse spatial state of the down-converted photons to be inferred. Although SLMs have previously been used to measure the orbital angular momentum of down-converted photons (*25*, *26*), we used a spatial variation in the blazing function (*27*) to also measure the angular position of the down-converted photons. The advantage of SLMs over static holographic components, or other phase filters, is that they can be electronically programmed to switch among different arbitrary measurement states.

To measure the orbital angular momentum state *28*) and single-photon measurement (*17*, *25*). The SLM is programmed with a spatial phase variation Φ(*x*, *y*), centered at *x* = 0, *y* = 0, given by*x*, *y*) given by*27*). The angle (arctan(*x*/*y*) – ϕ) is taken to be equal to the offset from the center of the aperture of width θ (we circumvented the numerical ambiguity in the vicinity of the angular origin). Although the Gaussian profile does correspond strictly to the minimum uncertainty state (*29*), the precise profile and minimum width of the sector apertures are not central to our argument nor to the results.

When either the signal or idler beam is viewed independently, the lack of constraint on its phase structure implies that a spatially coherent pump beam creates spatially incoherent down-converted light (*30*), with a speckle size dictated by the phase-matching bandwidth of the crystal. The size of this speckle with respect to the pump beam sets the modal spatial bandwidth of the incoherent output. For a plane-wave or Gaussian pump beam, conservation of orbital angular momentum between the signal and idler photons results in a high coincidence count rate whenever *31*) and further modified by the relative detection efficiency of the different states. Similarly, for measurements made in an image plane of the crystal the coincidence count rate for angular measurements is highest whenever ϕ* _{s}* – ϕ

*= 0.*

_{i}In practice, perfect correlations in both orbital angular momentum and angular position of the down-converted pairs are unobtainable. In order to demonstrate EPR correlations, we used the more experimentally useful criterion that is based on measuring the conditional probability of finding a particular outcome in one system given a measurement in the other (*32*). Angular momentum is discrete, and angular position is periodic. However, for narrow apertures the Fourier relationship between their variances is the same as for the unbounded continuous case (*22*), and we can write the EPR-Reid criterion as a violation of the inequality

For the orbital angular momentum states, we measured all combinations of * _{s}* and ϕ

*in 60 equally spaced angular bins. The measured correlations shown in Fig. 2, A and B, are maximal whenever*

_{i}*– ϕ*

_{s}*= 0, respectively. Shown in Fig. 2, C and D, are single sections through this data, illustrating the strength of the angular momentum and angle correlations.*

_{i}As with all experimental data, small amounts of random noise can adversely affect the statistical estimate of the variance of a distribution. We minimized this noise by running the experiment at a low flux (≈20,000 photon pairs s^{−1}), which minimizes the number of accidental coincidence counts arising from classical correlations.

Without any background subtraction, we obtained

We can also calculate the strength of the correlations when we subtract our best estimate of the background counts. For each measured number of coincidence counts *N*, we inferred the number of true coincidences *C* and accidental coincidences *B* so that *B* + *C* = *N*. Using the expected number of accidentals, calculated from *S _{s}S_{i}*Δ

*t*, the probability of a count being an accidental coincidence is taken to be

*r*= Min[1,

*S*Δ

_{s}S_{i}*t*/

*N*], so that the maximum probability is unity. The expectation value of

*B*is then

*Nr*. This enables a subtraction of accidental counts but ensures that we obtain nonnegative values for the calculation of probability or entropy.

Subtracting this calculated background to remove accidental coincidences, we obtained

We also adopted an alternative EPR criterion that is based on entropy rather than variance (*33*–*37*). This approach has a number of advantages. First, the entropic uncertainty relation does not require the calculation of a variance, and the calculation of entropy does not require us to deal with the complications associated with the cyclic nature of the angular variable (*38*, *39*). Secondly, the inequality is state-independent (*35*, *36*), which makes it possible to make a quantitative comparison of our EPR experiment with those reported for different physical systems. Finally, the entropic approach relates the strength of the correlations to the quantum information content of the system (*37*). The entropic uncertainty relation for angular position and angular momentum is (*35*, *36*)*N* segments filling the 2π interval. Direct comparison with our data, therefore, requires us to write our angle entropy in terms of the measured probabilities for these discrete segments ϕ* ^{m}* as (

*36*)

Using the data presented in Fig. 2, A and B, we obtained values of 1.548 ± 0.017 and 0.887 ± 0.018 for

The results confirm that the EPR conclusion, namely that quantum mechanics is incomplete or nonlocal, applies not only to position and momentum but also to angular position and angular momentum. Unlike demonstrations of Bell-type inequalities, which are restricted to discrete state spaces, EPR correlations simultaneously span an extended range of orbital angular momentum states and the continuous state space of angular position. The demonstration of angular EPR correlations establishes that angular position and angular momentum are suitable variables for applications in quantum information processing, notably in protocols for quantum key distribution (*40*).

## Supporting Online Material

www.sciencemag.org/cgi/content/full/329/5992/662/DC1

Materials and Methods

## References and Notes

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- This work is supported by the UK Engineering and Physical Sciences Research Council. S.F.A. is a Research Councils UK Research Fellow. S.M.B. and M.J.P. thank the Royal Society and the Wolfson Foundation. We thank J. Götte and D. Oi for useful discussions. We acknowledge the financial support of the Future and Emerging Technologies (FET) program within the Seventh Framework Programme for Research of the European Commission, under the FET Open grant agreement HIDEAS number FP7-ICT-221906. We would like to thank Hamamatsu for their support of this work. A.K.J. and R.W.B. were supported by a U.S. Department of Defense Multidisciplinary University Research Initiative award. The experiment was devised by J.L., M.J.P., and A.K.J. and performed by J.L., B.J., and J.R. D.I. designed the coincidence-counting electronics. S.M.B. and A.M.Y. formulated the inequalities. The results were interpreted and the manuscript drafted by M.J.P., S.M.B., A.M.Y., J.L., R.W.B., and S.F.A., with further inputs from all authors.