Abstract

Hardy’s nonlocality proof is considered as “the best version of Bell’s theorem”. We report an experimental implementation of this by measuring the orbital angular momentum (OAM) of entangled twisted photon pairs. Two advantages arise from using twisted photons. First, the limited OAM spectrum generated by parametric down-conversion provides a natural set of OAM non-maximally entangled states with selective degrees of entanglement. Second, the measurement of any non-trivial superposition of OAM states can be conveniently done with spatial light modulators. We measure states that are defined on asymmetric OAM Bloch spheres and show results which are incompatible with local realism.

© 2012 OSA

1. Introduction

In 1935 Einstein, Podolsky, and Rosen (EPR) raised a famous argument concerning the completeness of quantum mechanics [1]. Since then, there have been a number of schemes proposed to test local realism, which are generally grouped into two sets, namely, with and without inequalities [2]. Those with inequalities can be traced back to Bell’s theorem formulated in 1964 [3]. In contrast, Hardy in 1993 demonstrated a logical proof of nonlocality not involving inequalities [4] (see also [5, 6]), which was considered “the best version of Bell’s theorem” [7]. Hardy’s proof was first demonstrated by measuring photon polarization [8, 9]. A significant progress was the generalization to a ladder proof to increase the maximum fraction of measurements that demonstrate nonlocality [10, 11]. In this type of experiments, a source of non-maximally entangled photons is necessary. These sources have been implemented by clever arrangements of wave plates and Fresnel rhomb rotators to vary the degree of entanglement, and the correlations were inferred from Hong-Ou-Mandel peaks (dips) [10]. Another simpler implementation is a bidirectional-pump scheme for producing non-maximally entangled polar-isation states [12]. Here we report an experiment showing Hardy’s ladder proof, relying only on a standard parametric down-conversion setup without any sophisticated state preparation. Instead, we measure a different degree of freedom, namely the orbital angular momentum of entangled pairs of twisted photons.

Twisted photons are so named because of their characteristic helical wave fronts. OAM eigenstates |〉 have a phase front described by exp(iℓϕ), where ϕ is the azimuthal angle and is an integer. A photon in the state |〉 has an OAM of ℓh̄. Because is an integer, the OAM state-space is theoretically unbounded. This provides a promising playground for exploring high-dimensional entanglement and deeper features of quantum mechanics, and also for increasing the information capacity of photons [1315]. Our motivation for using OAM as another degree of freedom to test Hardy’s theory is twofold. First, there is a clear analogy between polarization and OAM, which allows for a straightforward implementation of the Hardy paradox in OAM. Second, spontaneous parametric downconversion (SPDC), which has proven to be a reliable source of entangled photons [16], result to a naturally nonmaximally entangled OAM state. The entanglement of OAM in these photon pairs has been established experimentally [17]. Since then, various tests of local realism, such as the EPR paradox [18] and Bell-type tests [19, 20] have been performed. While polarization is a two-dimensional state space, OAM is not [13]. We can choose to work on any two-dimensional subspace of the theoretically unbounded OAM state space. Choosing different subspaces effectively allows us to choose the degree of entanglement directly and without the introduction of any other component for state preparation. Using OAM states not only lead to convenient state preparation, but also to convenient state measurement when done with programmable spatial light modulators. The work we present here is the first time the Hardy paradox is demonstrated in the high-dimensional state-space of photon OAM.

2. Theory and experiment

Let us first summarize the Hardy paradox, in which quantum mechanics allows a set of probabilities that is logically inconsistent within a classical framework [47]. Assume there are two observers measuring dichotomic observables. Alice measures A0 and A1, while Bob measures B0 and B1. We define P(Ai, Bj) as the joint probability of obtaining Ai = 1 and Bj = 1, while P(Āi, Bj) is that of Ai = − 1 and Bj = 1. In the classical framework of local hidden-variable theory, if the three conditions: (I) P(A0, B0) = 0, (II) P(Ā0, B1) = 0 and (III) P(A1, 0) = 0 hold, then (IV) P(A1, B1) should be exactly zero. However, quantum mechanics allows suitable observables A0, B0 and A1, B1, satisfying (I), (II) and (III), but P1 = P(A1, B1) > 0. This has been generalised to a ladder proof [10], where when considering K + 1 defined dichotomic observables Ak and Bk where k = 0, 1,···,K), the following chain of probabilities hold:

P(AK,BK)>0,
P(A¯k1,Bk)=0,
P(Ak,B¯k1)=0,
P(A0,B0)=0.
As a result, the maximum Hardy fraction, PK = P(AK, BK) can be increased significantly. This chain of probabilities can only be shown for nonmaximally entangled states and is not valid for maximally entangled states [4]. Hence, it is essential to prepare nonmaximal polarization entanglement if one chooses to measure polarization, and this is often complicated.

SPDC naturally results to entangled OAM states which are not maximally entangled– the different OAM states have different weightings, and these weightings consist the spiral spectrum [21]. This suggests many sets of nonmaximally entangled states within different OAM subspaces with varying degrees of entanglement. We exploit the analogy between polarization and OAM. On the conventional Poincaré sphere [22], the north and south poles are left- and right- circular polarizations, respectively. The points on the equator indicate linear polarizations, the rest are elliptical polarizations. Although OAM has an unbounded number of orthogonal states, we can choose to work within two-dimensional subspaces represented by Bloch spheres [19]. Unlike [19], we construct asymmetric Bloch spheres spanned by OAM modes with different || values (e.g. Fig. 1). The north pole corresponds to | = +2〉 (or | = +1〉) and the south pole correspond to | = 0〉; the surface of each sphere encompasses all possible superpositions of these modes. The necessary measurements, {Ak} and {Bk}, to show the Hardy paradox reside on these spheres.

 

Fig. 1 The Bloch spheres for our Hardy test: (a) |+2〉 and |0〉; (b) |+1〉 and |0〉. We show the intensity of the modes on the poles and some representatives on the equator. We also show the holograms required for Alice’s measurements (Bob’s not shown).

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Our experiment is sketched in Fig. 2. A collimated 355 nm beam pumps a 5-mm long BBO crystal, where degenerate 710 nm signal and idler photons are produced in pairs via type-I collinear SPDC. These are separated by a non-polarizing beam splitter (BS). In each arm, we have a spatial light modulator (SLM) that allows us to program any phase profile. For the simple case of just measuring photons in OAM state |〉, the SLMs are encoded with a diffraction grating having dislocations. This simple hologram transforms light having the OAM state we intend to measure, into a fundamental mode in the direction of the first diffraction order. The SLMs are imaged onto single-mode fibers (SMF) and the output of the SMFs are connected to avalanche photodiodes whose outputs are fed to a coincidence counting circuit. The measurements needed to demonstrate the Hardy paradox consists of specific superpositions of OAM states which require modulation of both intensity and phase (Fig. 1). Nonetheless, we can employ the same concept to measure these states: we program our SLMs to holograms which consists of the specific superposition of OAM states, in addition to the grating. The SLM can only modulate the phase. However, we can incorporate a spatially dependent blazing function to the phase of the grating and the measurement state (Fig. 2), and this will allow us to also modulate the intensity [20]. This technique has been used previously for generating (in contrast to measuring) complex optical vortex topologies from the fundamental mode [23]. Intensity masking in this manner works, but there is a cost in efficiency. The holograms we use has a diffraction efficiency of about 60%, and the introduction of the blazing function will make this even smaller, translating to a decrease in coincidence counts.

 

Fig. 2 Experimental setup for showing Hardy’s paradox with entangled OAM. Left inset is spiral spectrum; top and bottom insets are examples of intensity and phase of measurement modes and the hologram displayed in SLM.

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We use the Laguerre-Gaussian modes characterised by the azimuthal index and radial index p, to express the state produced in SPDC [21, 24],

|ΨSPDC=ps,piCps,pi,|,ps|,pi,
where |Cps,pi,|2 is the probability of finding a signal photon in |, ps〉 and an idler photon in |−, pi〉. We restrict our measurements to the case where p = 0 and subsequently denote |, p = 0〉 as |〉. We first obtain the OAM entangled states available to us by measuring the spiral spectrum. We measure |〉 and | − 〉 using forked diffraction holograms [25], and build up the spiral spectrum from the coincidences measured for = −2 to +2, shown in the left inset of Fig. 2. We model the spiral spectrum as C0,0,=C=ζ||, where we find ζ = 0.66 ± 0.02 from an empirical fit. In reality, ζ is a function of the pump and detection waists, and the phase-matching conditions [24,25]. The important thing to note is that given an SPDC setup, the spiral spectrum can be obtained experimentally and this can be parametrized in terms of ||. This gives us a naturally non-maximally entangled two-photon state, and given the knowledge of the spiral spectrum we can choose different OAM subspaces with varying degrees of entanglement.

In our consideration, the Hilbert space of the signal photon is spanned by two arbitrary OAM eigenstates |mA and |nA. The idler space is then spanned by |− mB and |− nB, following the conservation law of angular momentum. Thus, the nonmaximally entangled state is naturally post-selected,

|Ψm,n=11+ε2(ε|mA|mB+|nA|nB),
where ε = Cm/Cnζ |m|−|n| denotes the degree of entanglement (ε = 1 for maximally entangled), depending on the chosen OAM bases. For the ladder version of Hardy’s paradox, let us define the following K + 1 OAM measurement bases, {Ak, Ak} and {Bk, Bk}, for signal and idler photons, respectively:
[|Ak|Ak]=[cosθkisinθkisinθkcosθk][|+mA|+nA],
[|Bk|Bk]=[cosθkisinθkisinθkcosθk][|mB|nB],

By substituting Eqs. (6)(8) to Eqs. (1)(4) and after some lengthy but straightforward algebra, we have tan θk = (−1)k × ζ(2k+1)(|m|−|n|)/2. Subsequently, we obtain the Hardy fraction,

PK=|AK|BK|Ψm,n|2=ε21+ε2(1ε2K1+ε2K+1)2.

In contrast to [10, 11], here the degree of entanglement is ε = 0.66|m|−|n| and can be easily varied by simply choosing different OAM bases, |m〉 and |n〉. The required measurements can be conveniently implemented using SLMs, which act as computer reconfigurable refractive elements and can be utilized to specify any non-trivial superpositions of OAM states [20]. Without loss of generality, we take two states, |Ψ〉2,0 and |Ψ〉1,0, for examples to demonstrate the Hardy paradox with K = 1 and K = 2, respectively. The corresponding degrees of entanglement, ε(2,0) = 0.43 and ε(1,0) = 0.66, are known from the measured spiral bandwidth (left inset in Fig. 2). Based on Eqs. (7)(9), we can calculate the states to be measured, and represent them on the asymmetric Bloch spheres, we show the holograms to measure the states |Ak〉 and |Ak (Fig. 1). We illustrate the case of K = 1 more closely for which we choose m = 2 and n = 0. In Fig. 2, we show the intensity and phase of the states we want to measure (|A0〉 in SLMA and |B0〉 in SLMB) and the holograms we use to detect these.

Our results are shown in Fig. 3(a) and 3(b). We obtain P1 = 0.0778 ± 0.0039 for |Ψ〉2,0 and P2 = 0.1389 ±0.0047 for |Ψ〉1,0, which are slightly less than the theoretical predictions of 0.0890 and 0.1573 from Eq. (10), respectively. The difference can be attributed to slight misalignment (the small, but nonzero probabilities for the rest measurements suggest some misalignment in our setup) and non-uniformity in the diffraction efficiency of different holograms used (we have been stringent in calculating the probabilities and we did not correct for these differences). All other probabilities are low as anticipated, consistent with the Hardy paradox. The Hardy paradox can also be put in a more general framework in terms of an inequality [10],

SK=P(AK,BK)P(A0,B0)k=1K[P(AK,B¯K1)+P(A¯K1,BK)]0,
which holds for any local hidden-variable theory. In contrast, here S1 = 0.069 ± 0.006 and S2 = 0.121±0.008, both evidently violating Eq. (10) and, therefore, contradicting local realism.

 

Fig. 3 Maximum fraction allowed by quantum mechanics (blank bars, red) and experimental results (solid, green) for (a) |Ψ〉2,0 with K = 1; (b) |Ψ〉1,0 with K = 2.

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3. Conclusion

To conclude, we have shown experimentally Hardy’s nonlocality proof by measuring the entangled OAM of twisted photon pairs. This enables us to directly use the nonmaximally entangled state produced by SPDC. We note that apart from post-selecting suitable OAM bases, we can also modify the degree of entanglement by tuning phase-matching conditions to directly change the spiral spectrum [25]. The good agreement between theoretical and experimental results point, again, to an incompatibility between local realism and quantum mechanics.

Acknowledgments

L.C. thanks the National Natural Science Foundation of China (NSFC) (grant 11104233), the Fundamental Research Funds for the Central Universities (grant 2011121043, 2012121015), and the Natural Science Foundation of Fujian Province of China (Grant No. 2011J05010). J.R thanks EPSRC. We thank Hamamatsu for support. L.C. and J.R. contributed equally to this work.

References and links

1. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of reality ever be considered complete?” Phys. Rev. 47, 777–780 (1935). [CrossRef]  

2. M. Genovese, “Research on hidden variable theories: a review of recent progresses,” Phys. Rep. 413, 319–396 (2005). [CrossRef]  

3. J. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys. 38, 447–452 (1966). [CrossRef]  

4. L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett. 71, 1665–1668 (1993). [CrossRef]  

5. S. Goldstein, “Nonlocality without inequalities for almost all entangled states for two particles,” Phys. Rev. Lett. 72, 1951–1951 (1994). [CrossRef]  

6. T. Jordan, “Testing einstein-podolsky-rosen assumptions without inequalities with two photons or particles with spin 1/2,” Phys. Rev. A 50, 62–66 (1994). [CrossRef]  

7. N. Mermin, “The best version of Bell’s theorem,” Ann. N. Y. Acad. Sci. 755, 616–623 (1995). [CrossRef]  

8. J. Torgerson, D. Branning, C. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A 204, 323–328 (1995). [CrossRef]  

9. G. Di Giuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A 56, 176–181 (1997). [CrossRef]  

10. D. Boschi, S. Branca, F. De Martini, and L. Hardy, “Ladder proof of nonlocality without inequalities: theoretical and experimental results,” Phys. Rev. Lett. 79, 2755–2758 (1997). [CrossRef]  

11. G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A 83, 042105 (2011). [CrossRef]  

12. M. Barbieri, F. De Martini, G. Di Nepi, and P. Mataloni, “Towards a test of non-locality without ’supplementary assumptions’,” Phys. Lett. A 334, 23–29 (2005). [CrossRef]  

13. G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [CrossRef]  

14. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2, 299–313 (2008). [CrossRef]  

15. M. Wiesniak, T. Paterek, and A. Zeilinger, “Entanglement in mutually unbiased bases,” New J. Phys. 13, 053047 (2011). [CrossRef]  

16. Y. Shih and C. Alley, “New type of eisntein-podolsky-rosen bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988). [CrossRef]  

17. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef]  

18. J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662 (2010). [CrossRef]  

19. J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 10, 8287–8293 (2009). [CrossRef]  

20. B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A 81, 43844 (2010). [CrossRef]  

21. J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003). [CrossRef]  

22. M. Padgett and J. Courtial, “Poincare-sphere equivalentfor light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999). [CrossRef]  

23. M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys. 6, 118–121 (2010). [CrossRef]  

24. A. Yao, “Spectral decomposition of entangled photons with an arbitrary pump,” New J. Phys. 13, 053048 (2011). [CrossRef]  

25. J. Romero, D. Giovannini, S. Franke-Anold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A 86, 012334 (2012). [CrossRef]  

References

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  1. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of reality ever be considered complete?” Phys. Rev.47, 777–780 (1935).
    [CrossRef]
  2. M. Genovese, “Research on hidden variable theories: a review of recent progresses,” Phys. Rep.413, 319–396 (2005).
    [CrossRef]
  3. J. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys.38, 447–452 (1966).
    [CrossRef]
  4. L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett.71, 1665–1668 (1993).
    [CrossRef]
  5. S. Goldstein, “Nonlocality without inequalities for almost all entangled states for two particles,” Phys. Rev. Lett.72, 1951–1951 (1994).
    [CrossRef]
  6. T. Jordan, “Testing einstein-podolsky-rosen assumptions without inequalities with two photons or particles with spin 1/2,” Phys. Rev. A50, 62–66 (1994).
    [CrossRef]
  7. N. Mermin, “The best version of Bell’s theorem,” Ann. N. Y. Acad. Sci.755, 616–623 (1995).
    [CrossRef]
  8. J. Torgerson, D. Branning, C. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A204, 323–328 (1995).
    [CrossRef]
  9. G. Di Giuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A56, 176–181 (1997).
    [CrossRef]
  10. D. Boschi, S. Branca, F. De Martini, and L. Hardy, “Ladder proof of nonlocality without inequalities: theoretical and experimental results,” Phys. Rev. Lett.79, 2755–2758 (1997).
    [CrossRef]
  11. G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
    [CrossRef]
  12. M. Barbieri, F. De Martini, G. Di Nepi, and P. Mataloni, “Towards a test of non-locality without ’supplementary assumptions’,” Phys. Lett. A334, 23–29 (2005).
    [CrossRef]
  13. G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys.3, 305–310 (2007).
    [CrossRef]
  14. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
    [CrossRef]
  15. M. Wiesniak, T. Paterek, and A. Zeilinger, “Entanglement in mutually unbiased bases,” New J. Phys.13, 053047 (2011).
    [CrossRef]
  16. Y. Shih and C. Alley, “New type of eisntein-podolsky-rosen bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
    [CrossRef]
  17. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001).
    [CrossRef]
  18. J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
    [CrossRef]
  19. J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
    [CrossRef]
  20. B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
    [CrossRef]
  21. J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A68, 050301 (2003).
    [CrossRef]
  22. M. Padgett and J. Courtial, “Poincare-sphere equivalentfor light beams containing orbital angular momentum,” Opt. Lett.24, 430–432 (1999).
    [CrossRef]
  23. M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys.6, 118–121 (2010).
    [CrossRef]
  24. A. Yao, “Spectral decomposition of entangled photons with an arbitrary pump,” New J. Phys.13, 053048 (2011).
    [CrossRef]
  25. J. Romero, D. Giovannini, S. Franke-Anold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86, 012334 (2012).
    [CrossRef]

2012 (1)

J. Romero, D. Giovannini, S. Franke-Anold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86, 012334 (2012).
[CrossRef]

2011 (3)

A. Yao, “Spectral decomposition of entangled photons with an arbitrary pump,” New J. Phys.13, 053048 (2011).
[CrossRef]

G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
[CrossRef]

M. Wiesniak, T. Paterek, and A. Zeilinger, “Entanglement in mutually unbiased bases,” New J. Phys.13, 053047 (2011).
[CrossRef]

2010 (3)

M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys.6, 118–121 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

2009 (1)

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

2008 (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

2007 (1)

G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys.3, 305–310 (2007).
[CrossRef]

2005 (2)

M. Barbieri, F. De Martini, G. Di Nepi, and P. Mataloni, “Towards a test of non-locality without ’supplementary assumptions’,” Phys. Lett. A334, 23–29 (2005).
[CrossRef]

M. Genovese, “Research on hidden variable theories: a review of recent progresses,” Phys. Rep.413, 319–396 (2005).
[CrossRef]

2003 (1)

J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A68, 050301 (2003).
[CrossRef]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001).
[CrossRef]

1999 (1)

1997 (2)

G. Di Giuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A56, 176–181 (1997).
[CrossRef]

D. Boschi, S. Branca, F. De Martini, and L. Hardy, “Ladder proof of nonlocality without inequalities: theoretical and experimental results,” Phys. Rev. Lett.79, 2755–2758 (1997).
[CrossRef]

1995 (2)

N. Mermin, “The best version of Bell’s theorem,” Ann. N. Y. Acad. Sci.755, 616–623 (1995).
[CrossRef]

J. Torgerson, D. Branning, C. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A204, 323–328 (1995).
[CrossRef]

1994 (2)

S. Goldstein, “Nonlocality without inequalities for almost all entangled states for two particles,” Phys. Rev. Lett.72, 1951–1951 (1994).
[CrossRef]

T. Jordan, “Testing einstein-podolsky-rosen assumptions without inequalities with two photons or particles with spin 1/2,” Phys. Rev. A50, 62–66 (1994).
[CrossRef]

1993 (1)

L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett.71, 1665–1668 (1993).
[CrossRef]

1988 (1)

Y. Shih and C. Alley, “New type of eisntein-podolsky-rosen bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
[CrossRef]

1966 (1)

J. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys.38, 447–452 (1966).
[CrossRef]

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of reality ever be considered complete?” Phys. Rev.47, 777–780 (1935).
[CrossRef]

Alexandrescu, A.

J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A68, 050301 (2003).
[CrossRef]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

Alley, C.

Y. Shih and C. Alley, “New type of eisntein-podolsky-rosen bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
[CrossRef]

Barbieri, M.

M. Barbieri, F. De Martini, G. Di Nepi, and P. Mataloni, “Towards a test of non-locality without ’supplementary assumptions’,” Phys. Lett. A334, 23–29 (2005).
[CrossRef]

Barnett, S.

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

Barnett, S. M.

J. Romero, D. Giovannini, S. Franke-Anold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86, 012334 (2012).
[CrossRef]

Bell, J.

J. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys.38, 447–452 (1966).
[CrossRef]

Boschi, D.

G. Di Giuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A56, 176–181 (1997).
[CrossRef]

D. Boschi, S. Branca, F. De Martini, and L. Hardy, “Ladder proof of nonlocality without inequalities: theoretical and experimental results,” Phys. Rev. Lett.79, 2755–2758 (1997).
[CrossRef]

Boyd, R.

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

Branca, S.

D. Boschi, S. Branca, F. De Martini, and L. Hardy, “Ladder proof of nonlocality without inequalities: theoretical and experimental results,” Phys. Rev. Lett.79, 2755–2758 (1997).
[CrossRef]

Branning, D.

J. Torgerson, D. Branning, C. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A204, 323–328 (1995).
[CrossRef]

Cabello, A.

G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
[CrossRef]

Courtial, J.

De Martini, F.

M. Barbieri, F. De Martini, G. Di Nepi, and P. Mataloni, “Towards a test of non-locality without ’supplementary assumptions’,” Phys. Lett. A334, 23–29 (2005).
[CrossRef]

G. Di Giuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A56, 176–181 (1997).
[CrossRef]

D. Boschi, S. Branca, F. De Martini, and L. Hardy, “Ladder proof of nonlocality without inequalities: theoretical and experimental results,” Phys. Rev. Lett.79, 2755–2758 (1997).
[CrossRef]

Dennis, M.

M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys.6, 118–121 (2010).
[CrossRef]

Di Giuseppe, G.

G. Di Giuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A56, 176–181 (1997).
[CrossRef]

Di Nepi, G.

M. Barbieri, F. De Martini, G. Di Nepi, and P. Mataloni, “Towards a test of non-locality without ’supplementary assumptions’,” Phys. Lett. A334, 23–29 (2005).
[CrossRef]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of reality ever be considered complete?” Phys. Rev.47, 777–780 (1935).
[CrossRef]

Franke-Anold, S.

J. Romero, D. Giovannini, S. Franke-Anold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86, 012334 (2012).
[CrossRef]

Franke-Arnold, S.

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

Genovese, M.

M. Genovese, “Research on hidden variable theories: a review of recent progresses,” Phys. Rep.413, 319–396 (2005).
[CrossRef]

Gianani, I.

G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
[CrossRef]

Giovannini, D.

J. Romero, D. Giovannini, S. Franke-Anold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86, 012334 (2012).
[CrossRef]

Goldstein, S.

S. Goldstein, “Nonlocality without inequalities for almost all entangled states for two particles,” Phys. Rev. Lett.72, 1951–1951 (1994).
[CrossRef]

Hardy, L.

D. Boschi, S. Branca, F. De Martini, and L. Hardy, “Ladder proof of nonlocality without inequalities: theoretical and experimental results,” Phys. Rev. Lett.79, 2755–2758 (1997).
[CrossRef]

L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett.71, 1665–1668 (1993).
[CrossRef]

Inostroza, E.

G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
[CrossRef]

Ireland, D.

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

Jack, B.

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys.6, 118–121 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

Jha, A.

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

Jordan, T.

T. Jordan, “Testing einstein-podolsky-rosen assumptions without inequalities with two photons or particles with spin 1/2,” Phys. Rev. A50, 62–66 (1994).
[CrossRef]

King, R.P.

M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys.6, 118–121 (2010).
[CrossRef]

Leach, J.

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

Lima, G.

G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001).
[CrossRef]

Mandel, L.

J. Torgerson, D. Branning, C. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A204, 323–328 (1995).
[CrossRef]

Mataloni, P.

G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
[CrossRef]

M. Barbieri, F. De Martini, G. Di Nepi, and P. Mataloni, “Towards a test of non-locality without ’supplementary assumptions’,” Phys. Lett. A334, 23–29 (2005).
[CrossRef]

Mermin, N.

N. Mermin, “The best version of Bell’s theorem,” Ann. N. Y. Acad. Sci.755, 616–623 (1995).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys.3, 305–310 (2007).
[CrossRef]

Monken, C.

J. Torgerson, D. Branning, C. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A204, 323–328 (1995).
[CrossRef]

O’Holleran, K.

M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys.6, 118–121 (2010).
[CrossRef]

Padgett, M.

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

M. Padgett and J. Courtial, “Poincare-sphere equivalentfor light beams containing orbital angular momentum,” Opt. Lett.24, 430–432 (1999).
[CrossRef]

Padgett, M. J.

J. Romero, D. Giovannini, S. Franke-Anold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86, 012334 (2012).
[CrossRef]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

Padgett, M.J.

M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys.6, 118–121 (2010).
[CrossRef]

Paterek, T.

M. Wiesniak, T. Paterek, and A. Zeilinger, “Entanglement in mutually unbiased bases,” New J. Phys.13, 053047 (2011).
[CrossRef]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of reality ever be considered complete?” Phys. Rev.47, 777–780 (1935).
[CrossRef]

Ritsch-Marte, M.

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

Romero, J.

J. Romero, D. Giovannini, S. Franke-Anold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86, 012334 (2012).
[CrossRef]

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of reality ever be considered complete?” Phys. Rev.47, 777–780 (1935).
[CrossRef]

Saavedra, C.

G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
[CrossRef]

Shih, Y.

Y. Shih and C. Alley, “New type of eisntein-podolsky-rosen bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
[CrossRef]

Torgerson, J.

J. Torgerson, D. Branning, C. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A204, 323–328 (1995).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys.3, 305–310 (2007).
[CrossRef]

J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A68, 050301 (2003).
[CrossRef]

Torres, J.

G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys.3, 305–310 (2007).
[CrossRef]

J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A68, 050301 (2003).
[CrossRef]

Vallone, G.

G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
[CrossRef]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001).
[CrossRef]

Wiesniak, M.

M. Wiesniak, T. Paterek, and A. Zeilinger, “Entanglement in mutually unbiased bases,” New J. Phys.13, 053047 (2011).
[CrossRef]

Yao, A.

A. Yao, “Spectral decomposition of entangled photons with an arbitrary pump,” New J. Phys.13, 053048 (2011).
[CrossRef]

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

Zeilinger, A.

M. Wiesniak, T. Paterek, and A. Zeilinger, “Entanglement in mutually unbiased bases,” New J. Phys.13, 053047 (2011).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001).
[CrossRef]

Ann. N. Y. Acad. Sci. (1)

N. Mermin, “The best version of Bell’s theorem,” Ann. N. Y. Acad. Sci.755, 616–623 (1995).
[CrossRef]

Laser Photonics Rev. (1)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008).
[CrossRef]

Nat. Phys. (2)

G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys.3, 305–310 (2007).
[CrossRef]

M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys.6, 118–121 (2010).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001).
[CrossRef]

New J. Phys. (2)

M. Wiesniak, T. Paterek, and A. Zeilinger, “Entanglement in mutually unbiased bases,” New J. Phys.13, 053047 (2011).
[CrossRef]

A. Yao, “Spectral decomposition of entangled photons with an arbitrary pump,” New J. Phys.13, 053048 (2011).
[CrossRef]

Opt. Express (1)

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express10, 8287–8293 (2009).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. A (2)

M. Barbieri, F. De Martini, G. Di Nepi, and P. Mataloni, “Towards a test of non-locality without ’supplementary assumptions’,” Phys. Lett. A334, 23–29 (2005).
[CrossRef]

J. Torgerson, D. Branning, C. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A204, 323–328 (1995).
[CrossRef]

Phys. Rep. (1)

M. Genovese, “Research on hidden variable theories: a review of recent progresses,” Phys. Rep.413, 319–396 (2005).
[CrossRef]

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of reality ever be considered complete?” Phys. Rev.47, 777–780 (1935).
[CrossRef]

Phys. Rev. A (6)

T. Jordan, “Testing einstein-podolsky-rosen assumptions without inequalities with two photons or particles with spin 1/2,” Phys. Rev. A50, 62–66 (1994).
[CrossRef]

G. Vallone, I. Gianani, E. Inostroza, C. Saavedra, G. Lima, A. Cabello, and P. Mataloni, “Testing hardys nonlocality proof with genuine energy-time entanglement,” Phys. Rev. A83, 042105 (2011).
[CrossRef]

G. Di Giuseppe, F. De Martini, and D. Boschi, “Experimental test of the violation of local realism in quantum mechanics without Bell inequalities,” Phys. Rev. A56, 176–181 (1997).
[CrossRef]

B. Jack, A. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. Ireland, S. Barnett, and M. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A81, 43844 (2010).
[CrossRef]

J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A68, 050301 (2003).
[CrossRef]

J. Romero, D. Giovannini, S. Franke-Anold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A86, 012334 (2012).
[CrossRef]

Phys. Rev. Lett. (4)

D. Boschi, S. Branca, F. De Martini, and L. Hardy, “Ladder proof of nonlocality without inequalities: theoretical and experimental results,” Phys. Rev. Lett.79, 2755–2758 (1997).
[CrossRef]

L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett.71, 1665–1668 (1993).
[CrossRef]

S. Goldstein, “Nonlocality without inequalities for almost all entangled states for two particles,” Phys. Rev. Lett.72, 1951–1951 (1994).
[CrossRef]

Y. Shih and C. Alley, “New type of eisntein-podolsky-rosen bohm experiment using pairs of light quanta produced by optical parametric down conversion,” Phys. Rev. Lett.61, 2921–2924 (1988).
[CrossRef]

Rev. Mod. Phys. (1)

J. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys.38, 447–452 (1966).
[CrossRef]

Science (1)

J. Leach, B. Jack, J. Romero, A. Jha, A. Yao, S. Franke-Arnold, D. Ireland, R. Boyd, S. Barnett, and M. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science329, 662 (2010).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

The Bloch spheres for our Hardy test: (a) |+2〉 and |0〉; (b) |+1〉 and |0〉. We show the intensity of the modes on the poles and some representatives on the equator. We also show the holograms required for Alice’s measurements (Bob’s not shown).

Fig. 2
Fig. 2

Experimental setup for showing Hardy’s paradox with entangled OAM. Left inset is spiral spectrum; top and bottom insets are examples of intensity and phase of measurement modes and the hologram displayed in SLM.

Fig. 3
Fig. 3

Maximum fraction allowed by quantum mechanics (blank bars, red) and experimental results (solid, green) for (a) |Ψ〉2,0 with K = 1; (b) |Ψ〉1,0 with K = 2.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

P ( A K , B K ) > 0 ,
P ( A ¯ k 1 , B k ) = 0 ,
P ( A k , B ¯ k 1 ) = 0 ,
P ( A 0 , B 0 ) = 0.
| Ψ S P D C = p s , p i C p s , p i , | , p s | , p i ,
| Ψ m , n = 1 1 + ε 2 ( ε | m A | m B + | n A | n B ) ,
[ | A k | A k ] = [ cos θ k i sin θ k i sin θ k cos θ k ] [ | + m A | + n A ] ,
[ | B k | B k ] = [ cos θ k i sin θ k i sin θ k cos θ k ] [ | m B | n B ] ,
P K = | A K | B K | Ψ m , n | 2 = ε 2 1 + ε 2 ( 1 ε 2 K 1 + ε 2 K + 1 ) 2 .
S K = P ( A K , B K ) P ( A 0 , B 0 ) k = 1 K [ P ( A K , B ¯ K 1 ) + P ( A ¯ K 1 , B K ) ] 0 ,

Metrics