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

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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]

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[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)

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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 ,

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