## Hardy’s nonlocality proof using twisted photons |

Optics Express, Vol. 20, Issue 19, pp. 21687-21692 (2012)

http://dx.doi.org/10.1364/OE.20.021687

Acrobat PDF (829 KB)

### 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

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]

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]

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]

*ℓ*〉 have a phase front described by exp(

*iℓϕ*), where

*ϕ*is the azimuthal angle and

*ℓ*is an integer. A photon in the state |

*ℓ*〉 has an OAM of

*ℓh̄*. Because

*ℓ*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 [13

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

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

## 2. Theory and experiment

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]

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]

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]

*ℓ*| values (e.g. Fig. 1). The north pole corresponds to |

*ℓ*= +2〉 (or |

*ℓ*= +1〉) and the south pole correspond to |

*ℓ*= 0〉; the surface of each sphere encompasses all possible superpositions of these modes. The necessary measurements, {

*A*} and {

_{k}*B*}, to show the Hardy paradox reside on these spheres.

_{k}*ℓ*〉, the SLMs are encoded with a diffraction grating having

*ℓ*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

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]

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]

*ℓ*and radial index

*p*, to express the state produced in SPDC [21

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

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

*ℓ*,

*p*〉 and an idler photon in |−

_{s}*ℓ*,

*p*〉. We restrict our measurements to the case where

_{i}*p*= 0 and subsequently denote |

*ℓ*,

*p*= 0〉 as |

*ℓ*〉. We first obtain the OAM entangled states available to us by measuring the spiral spectrum. We measure |

*ℓ*〉 and | −

*ℓ*〉 using forked diffraction holograms [25

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]

*ℓ*= −2 to +2, shown in the left inset of Fig. 2. We model the spiral spectrum as

*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

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]

*ℓ*|. 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.

*m*〉

*and |*

_{A}*n*〉

*. The idler space is then spanned by |−*

_{A}*m*〉

*and |−*

_{B}*n*〉

*, following the conservation law of angular momentum. Thus, the nonmaximally entangled state is naturally post-selected, where*

_{B}*ε*=

*C*/

_{m}*C*≈

_{n}*ζ*

^{|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, {

*A*,

_{k}*B*,

_{k}*θ*= (−1)

_{k}*×*

^{k}*ζ*

^{(2k+1)(|m|−|n|)/2}. Subsequently, we obtain the Hardy fraction,

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]

*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

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]

_{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 |

*A*〉 and

_{k}*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 (|

*A*

_{0}〉 in SLM

*and |*

_{A}*B*

_{0}〉 in SLM

*) and the holograms we use to detect these.*

_{B}## 3. Conclusion

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]

## Acknowledgments

## References and links

1. | A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of reality ever be considered complete?” Phys. Rev. |

2. | M. Genovese, “Research on hidden variable theories: a review of recent progresses,” Phys. Rep. |

3. | J. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys. |

4. | L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett. |

5. | S. Goldstein, “Nonlocality without inequalities for almost all entangled states for two particles,” Phys. Rev. Lett. |

6. | T. Jordan, “Testing einstein-podolsky-rosen assumptions without inequalities with two photons or particles with spin 1/2,” Phys. Rev. A |

7. | N. Mermin, “The best version of Bell’s theorem,” Ann. N. Y. Acad. Sci. |

8. | J. Torgerson, D. Branning, C. Monken, and L. Mandel, “Experimental demonstration of the violation of local realism without Bell inequalities,” Phys. Lett. A |

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 |

10. | D. Boschi, S. Branca, F. De Martini, and L. Hardy, “Ladder proof of nonlocality without inequalities: theoretical and experimental results,” Phys. Rev. Lett. |

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 |

12. | M. Barbieri, F. De Martini, G. Di Nepi, and P. Mataloni, “Towards a test of non-locality without ’supplementary assumptions’,” Phys. Lett. A |

13. | G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys. |

14. | S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. |

15. | M. Wiesniak, T. Paterek, and A. Zeilinger, “Entanglement in mutually unbiased bases,” New J. Phys. |

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. |

17. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

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 |

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 |

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 |

21. | J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A |

22. | M. Padgett and J. Courtial, “Poincare-sphere equivalentfor light beams containing orbital angular momentum,” Opt. Lett. |

23. | M. Dennis, R.P. King, B. Jack, K. O’Holleran, and M.J. Padgett, “Isolated optical vortex knots”, Nat. Phys. |

24. | A. Yao, “Spectral decomposition of entangled photons with an arbitrary pump,” New J. Phys. |

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 |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(050.4865) Diffraction and gratings : Optical vortices

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: June 28, 2012

Revised Manuscript: August 28, 2012

Manuscript Accepted: August 31, 2012

Published: September 6, 2012

**Citation**

Lixiang Chen and Jacquiline Romero, "Hardy’s nonlocality proof using twisted photons," Opt. Express **20**, 21687-21692 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-21687

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### References

- A. Einstein, B. Podolsky, and N. Rosen, “Can quantum mechanical description of reality ever be considered complete?” Phys. Rev.47, 777–780 (1935). [CrossRef]
- M. Genovese, “Research on hidden variable theories: a review of recent progresses,” Phys. Rep.413, 319–396 (2005). [CrossRef]
- J. Bell, “On the problem of hidden variables in quantum mechanics,” Rev. Mod. Phys.38, 447–452 (1966). [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]
- 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]
- 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]
- 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]
- 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]
- G. Molina-Terriza, J. Torres, and L. Torner, “Twisted photons,” Nat. Phys.3, 305–310 (2007). [CrossRef]
- S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008). [CrossRef]
- M. Wiesniak, T. Paterek, and A. Zeilinger, “Entanglement in mutually unbiased bases,” New J. Phys.13, 053047 (2011). [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]
- A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412, 313–316 (2001). [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]
- 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]
- M. Padgett and J. Courtial, “Poincare-sphere equivalentfor light beams containing orbital angular momentum,” Opt. Lett.24, 430–432 (1999). [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]
- A. Yao, “Spectral decomposition of entangled photons with an arbitrary pump,” New J. Phys.13, 053048 (2011). [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]

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