Abstract

Using angular position–orbital angular momentum entangled photons, we propose an experiment to generate maximally entangled states in D-dimensional quantum systems, the so called qudits, by exploiting correlations of parametric down-converted photons. Angular diffraction masks containing N angular slits in the arms of each twin photon define a qudit space of dimension N2, spanned by the alternative pathways of the photons. Numerical results for N angular slits with N = 2, 4, 5, 10 are reported. We discuss relevant experimental parameters for an experimental implementation of the proposed scheme using Spatial Light Modulators (SLMs), and twin-photons produced by Spontaneous Parametric Down Conversion (SPDC). The entanglement of the qudit state can be quantified in terms of the Concurrence, which can be expressed in terms of the visibility of the interference fringes, or by using Entanglement Witnesses. These results provide an additional means for preparing entangled quantum states in high-dimensions, a fundamental resource for quantum simulation and quantum information protocols.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

G. Sorelli, V. N Shatokhin, and A. Buchleitner, “Universal entanglement loss induced by angular uncertainty,” J. Opt. 22(2), 024002 (2020).
[Crossref]

2019 (1)

P. Machado, A. A. Matoso, M. R. Barros, L. Neves, and S. Padua, “Engineering quantum correlations for m×n spatially encoded two-photons states,” Phys. Rev. A 99(6), 063839 (2019).
[Crossref]

2018 (1)

2015 (1)

G. Puentes, G. Waldherr, P. Neumann, G. Balasubramanian, and J. Wrachtrup, “Efficient route to high-bandwidth nanoscale magnetometry using single spins in diamond,” Sci. Rep. 4(1), 4677 (2015).
[Crossref]

2013 (2)

G. Puentes, G. Colangelo, R. J. Sewell, and M. W. Mitchell, “Planar squeezing by quantum non-demolition measurement in cold atomic ensembles,” New J. Phys. 15(10), 103031 (2013).
[Crossref]

S. Moulieras, M. Lewenstein, and G. Puentes, “Entanglement engineering and topological protection by discrete-time quantum walks,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104005 (2013).
[Crossref]

2011 (1)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalised Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

2010 (2)

A. Kumar Jha, J. Leach, B. Jack, S. Franke-Arnold, S. Barnett, R. Boyd, and M. Padgett, “Angular Two-Photon Interference and Angular Two-Qubit States,” Phys. Rev. Lett. 104(1), 010501 (2010).
[Crossref]

G. Puentes, A. Datta, A. Feito, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Entanglement quantification from incomplete measurements: applications using photon-number-resolving weak homodyne detectors,” New J. Phys. 12(3), 033042 (2010).
[Crossref]

2009 (3)

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete Tunable Color Entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009).
[Crossref]

B. Jack, J. Leach, H. Ritsch, S. M. Barnet, M. J. Padgett, and S. Franke-Arnold, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 11(10), 103024 (2009).
[Crossref]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. W. Boyd, A. K. Jha, and S. M. Barnett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17(10), 8287 (2009).
[Crossref]

2008 (3)

K. Jha, M. N. O’Sullivan, K. W. Clifford Chan, and R. W. Boyd, “Temporal coherence and indistinguishability in two-photon interference effects,” Phys. Rev. A 77(2), 021801 (2008).
[Crossref]

B. Jack, M. Padgett, and S. Franke-Arnold, “Angular Diffraction,” New J. Phys. 10(10), 103013 (2008).
[Crossref]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

2007 (3)

N. Leonardo, G. Lima, E. J. S. Fonseca, L. Davidovich, and S. Pádua, “Characterizing entanglement in qubits created with spatially correlated twin photons,” Phys. Rev. A 76(3), 032314 (2007).
[Crossref]

G. Puentes, A. Aiello, D. Voigt, and J. P. Woerdman, “Maximally entangled mixed-state generation via local operations,” Phys. Rev. A 75(3), 032319 (2007).
[Crossref]

J. Eisert, F. G. S. L. Brandão, and K. M. R. Audenaert, “Quantitative entanglement witnesses,” New J. Phys. 9(3), 46 (2007).
[Crossref]

2006 (3)

G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Tunable spatial decoherers for polarization-entangled photons,” Opt. Lett. 31(13), 2057–2059 (2006).
[Crossref]

A. Ling, P. Y. Han, A. Lamas-Limares, and C. Kurtsiefer, “Preparation of Bell state with controlled white noise,” Laser Phys. 16(7), 1140–1144 (2006).
[Crossref]

K. M. R. Audenaert and M. B. Plenio, “When are correlations quantum?—verification and quantification of entanglement by simple measurements,” New J. Phys. 8(11), 266 (2006).
[Crossref]

2005 (4)

T-C Wei, J. B. Altepeter, D. Branning, P. M. Goldbart, D. F. James, E. Jeffrey, P. G. Kwiat, S. Mukhopadhyay, and N. Peters, “Synthesizing arbitrary two-photon mixed states,” Phys. Rev. A 71(3), 032329 (2005).
[Crossref]

M. N. O’Sullivan-Hale, I. A. Khan, Robert W. Boyd, and John C. Howell, “Pixel Entanglement: Experimental Realization of Optically Entangled d=3 and d=6 Qudits,” Phys. Rev. Lett. 94(22), 220501 (2005).
[Crossref]

N. Leonardo, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of Entangled States of Qudits using Twin Photons,” Phys. Rev. Lett. 94(10), 100501 (2005).
[Crossref]

M. B. Plenio, “Logarithmic Negativity: A Full Entanglement Monotone That is not Convex,” Phys. Rev. Lett. 95(9), 090503 (2005).
[Crossref]

2004 (2)

R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, “Bell-Type Test of Energy-Time Entangled Qutrits,” Phys. Rev. Lett. 93(1), 010503 (2004).
[Crossref]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

2003 (1)

T. Durt, N. J. Cerf, N. Gisin, and M. Zukowski, “Security of quantum key distribution with entangled qutrits,” Phys. Rev. A 67(1), 012311 (2003).
[Crossref]

2002 (6)

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of Quantum Key Distribution Using d-Level Systems,” Phys. Rev. Lett. 88(12), 127902 (2002).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref]

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell Inequalities for Arbitrarily High-Dimensional Systems,” Phys. Rev. Lett. 88(4), 040404 (2002).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref]

R. Thew, S. Tanzilli W. Tittel, H. Zbinden, and N. Gisin, “Experimental investigation of the robustness of partially entangled qubits over 11 km,” Phys. Rev. A 66(6), 062304 (2002).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref]

2001 (2)

P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions,” Phys. Rev. A 64(4), 042315 (2001).
[Crossref]

M. Bourennane, A. Karlsson, and G. Bjork, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

2000 (3)

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of Local Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubits,” Phys. Rev. Lett. 85(21), 4418–4421 (2000).
[Crossref]

H. Bechmann-Pasquinucci and A. Peres, “Quantum Cryptography with 3-State Systems,” Phys. Rev. Lett. 85(15), 3313–3316 (2000).
[Crossref]

E. J. S. Fonseca, P. H. Souto Ribeiro, S. Padua, and C. H. Monken, “Quantum interference by a nonlocal double slit,” Phys. Rev. A 61(2), 023801 (2000).
[Crossref]

1999 (2)

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication,” Phys. Rev. Lett. 82(12), 2594–2597 (1999).
[Crossref]

A. Zeilinger, “Experiment and the foundations of quantum physics,” Rev. Mod. Phys. 71(2), S288–S297 (1999).
[Crossref]

1998 (2)

1995 (1)

P. G. Mattle, K. Weinfurter, H. Zeilinger, A. Sergienko, A. V. Shih, and Y. Kwiat, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995).
[Crossref]

1994 (1)

T. J. Herzog, J. G. Rarity, H. Weinfurter, and A. Zeilinger, “Frustrated two-photon creation via interference,” Phys. Rev. Lett. 72(5), 629–632 (1994).
[Crossref]

1993 (1)

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, and A. Peres, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895–1899 (1993).
[Crossref]

1992 (1)

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69(20), 2881–2884 (1992).
[Crossref]

1991 (2)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref]

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref]

1990 (2)

S. M. Barnett and D. T. Pegg, “Quantum theory of rotation angles,” Phys. Rev. A 41(7), 3427–3435 (1990).
[Crossref]

J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64(21), 2495–2498 (1990).
[Crossref]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref]

1982 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell’s Inequalities,” Phys. Rev. Lett. 49(2), 91–94 (1982).
[Crossref]

Acin, A.

R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, “Bell-Type Test of Energy-Time Entangled Qutrits,” Phys. Rev. Lett. 93(1), 010503 (2004).
[Crossref]

Aguirre Gómez, J. G.

N. Leonardo, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of Entangled States of Qudits using Twin Photons,” Phys. Rev. Lett. 94(10), 100501 (2005).
[Crossref]

Aiello, A.

G. Puentes, A. Aiello, D. Voigt, and J. P. Woerdman, “Maximally entangled mixed-state generation via local operations,” Phys. Rev. A 75(3), 032319 (2007).
[Crossref]

G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Tunable spatial decoherers for polarization-entangled photons,” Opt. Lett. 31(13), 2057–2059 (2006).
[Crossref]

Alexander, V. Sergienko

Altepeter, J. B.

T-C Wei, J. B. Altepeter, D. Branning, P. M. Goldbart, D. F. James, E. Jeffrey, P. G. Kwiat, S. Mukhopadhyay, and N. Peters, “Synthesizing arbitrary two-photon mixed states,” Phys. Rev. A 71(3), 032329 (2005).
[Crossref]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalised Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

Aspect, A.

A. Aspect, P. Grangier, and G. Roger, “Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell’s Inequalities,” Phys. Rev. Lett. 49(2), 91–94 (1982).
[Crossref]

Audenaert, K. M. R.

J. Eisert, F. G. S. L. Brandão, and K. M. R. Audenaert, “Quantitative entanglement witnesses,” New J. Phys. 9(3), 46 (2007).
[Crossref]

K. M. R. Audenaert and M. B. Plenio, “When are correlations quantum?—verification and quantification of entanglement by simple measurements,” New J. Phys. 8(11), 266 (2006).
[Crossref]

Ayman, F. Abouraddy

Balasubramanian, G.

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P. G. Mattle, K. Weinfurter, H. Zeilinger, A. Sergienko, A. V. Shih, and Y. Kwiat, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995).
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D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of Local Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubits,” Phys. Rev. Lett. 85(21), 4418–4421 (2000).
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G. Puentes, G. Colangelo, R. J. Sewell, and M. W. Mitchell, “Planar squeezing by quantum non-demolition measurement in cold atomic ensembles,” New J. Phys. 15(10), 103031 (2013).
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N. Leonardo, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of Entangled States of Qudits using Twin Photons,” Phys. Rev. Lett. 94(10), 100501 (2005).
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E. J. S. Fonseca, P. H. Souto Ribeiro, S. Padua, and C. H. Monken, “Quantum interference by a nonlocal double slit,” Phys. Rev. A 61(2), 023801 (2000).
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S. Moulieras, M. Lewenstein, and G. Puentes, “Entanglement engineering and topological protection by discrete-time quantum walks,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104005 (2013).
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T-C Wei, J. B. Altepeter, D. Branning, P. M. Goldbart, D. F. James, E. Jeffrey, P. G. Kwiat, S. Mukhopadhyay, and N. Peters, “Synthesizing arbitrary two-photon mixed states,” Phys. Rev. A 71(3), 032329 (2005).
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G. Puentes, G. Waldherr, P. Neumann, G. Balasubramanian, and J. Wrachtrup, “Efficient route to high-bandwidth nanoscale magnetometry using single spins in diamond,” Sci. Rep. 4(1), 4677 (2015).
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P. Machado, A. A. Matoso, M. R. Barros, L. Neves, and S. Padua, “Engineering quantum correlations for m×n spatially encoded two-photons states,” Phys. Rev. A 99(6), 063839 (2019).
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K. Jha, M. N. O’Sullivan, K. W. Clifford Chan, and R. W. Boyd, “Temporal coherence and indistinguishability in two-photon interference effects,” Phys. Rev. A 77(2), 021801 (2008).
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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalised Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
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P. Machado, A. A. Matoso, M. R. Barros, L. Neves, and S. Padua, “Engineering quantum correlations for m×n spatially encoded two-photons states,” Phys. Rev. A 99(6), 063839 (2019).
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G. Puentes, A. Datta, A. Feito, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Entanglement quantification from incomplete measurements: applications using photon-number-resolving weak homodyne detectors,” New J. Phys. 12(3), 033042 (2010).
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G. Puentes, G. Colangelo, R. J. Sewell, and M. W. Mitchell, “Planar squeezing by quantum non-demolition measurement in cold atomic ensembles,” New J. Phys. 15(10), 103031 (2013).
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S. Moulieras, M. Lewenstein, and G. Puentes, “Entanglement engineering and topological protection by discrete-time quantum walks,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104005 (2013).
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G. Puentes, A. Datta, A. Feito, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Entanglement quantification from incomplete measurements: applications using photon-number-resolving weak homodyne detectors,” New J. Phys. 12(3), 033042 (2010).
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G. Puentes, A. Aiello, D. Voigt, and J. P. Woerdman, “Maximally entangled mixed-state generation via local operations,” Phys. Rev. A 75(3), 032319 (2007).
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G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Tunable spatial decoherers for polarization-entangled photons,” Opt. Lett. 31(13), 2057–2059 (2006).
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S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete Tunable Color Entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009).
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B. Jack, J. Leach, H. Ritsch, S. M. Barnet, M. J. Padgett, and S. Franke-Arnold, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 11(10), 103024 (2009).
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P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions,” Phys. Rev. A 64(4), 042315 (2001).
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Saavedra, C.

N. Leonardo, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of Entangled States of Qudits using Twin Photons,” Phys. Rev. Lett. 94(10), 100501 (2005).
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Saleh, Bahaa E. A.

Sergienko, A.

P. G. Mattle, K. Weinfurter, H. Zeilinger, A. Sergienko, A. V. Shih, and Y. Kwiat, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995).
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G. Puentes, G. Colangelo, R. J. Sewell, and M. W. Mitchell, “Planar squeezing by quantum non-demolition measurement in cold atomic ensembles,” New J. Phys. 15(10), 103031 (2013).
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G. Sorelli, V. N Shatokhin, and A. Buchleitner, “Universal entanglement loss induced by angular uncertainty,” J. Opt. 22(2), 024002 (2020).
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P. G. Mattle, K. Weinfurter, H. Zeilinger, A. Sergienko, A. V. Shih, and Y. Kwiat, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995).
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G. Sorelli, V. N Shatokhin, and A. Buchleitner, “Universal entanglement loss induced by angular uncertainty,” J. Opt. 22(2), 024002 (2020).
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E. J. S. Fonseca, P. H. Souto Ribeiro, S. Padua, and C. H. Monken, “Quantum interference by a nonlocal double slit,” Phys. Rev. A 61(2), 023801 (2000).
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Takayama, O.

Tanzilli W. Tittel, S.

R. Thew, S. Tanzilli W. Tittel, H. Zbinden, and N. Gisin, “Experimental investigation of the robustness of partially entangled qubits over 11 km,” Phys. Rev. A 66(6), 062304 (2002).
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J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64(21), 2495–2498 (1990).
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R. Thew, S. Tanzilli W. Tittel, H. Zbinden, and N. Gisin, “Experimental investigation of the robustness of partially entangled qubits over 11 km,” Phys. Rev. A 66(6), 062304 (2002).
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R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, “Bell-Type Test of Energy-Time Entangled Qutrits,” Phys. Rev. Lett. 93(1), 010503 (2004).
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J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication,” Phys. Rev. Lett. 82(12), 2594–2597 (1999).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
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Voigt, D.

G. Puentes, A. Aiello, D. Voigt, and J. P. Woerdman, “Maximally entangled mixed-state generation via local operations,” Phys. Rev. A 75(3), 032319 (2007).
[Crossref]

G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Tunable spatial decoherers for polarization-entangled photons,” Opt. Lett. 31(13), 2057–2059 (2006).
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Waldherr, G.

G. Puentes, G. Waldherr, P. Neumann, G. Balasubramanian, and J. Wrachtrup, “Efficient route to high-bandwidth nanoscale magnetometry using single spins in diamond,” Sci. Rep. 4(1), 4677 (2015).
[Crossref]

Walmsley, I. A.

G. Puentes, A. Datta, A. Feito, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Entanglement quantification from incomplete measurements: applications using photon-number-resolving weak homodyne detectors,” New J. Phys. 12(3), 033042 (2010).
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T-C Wei, J. B. Altepeter, D. Branning, P. M. Goldbart, D. F. James, E. Jeffrey, P. G. Kwiat, S. Mukhopadhyay, and N. Peters, “Synthesizing arbitrary two-photon mixed states,” Phys. Rev. A 71(3), 032329 (2005).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
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A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
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T. J. Herzog, J. G. Rarity, H. Weinfurter, and A. Zeilinger, “Frustrated two-photon creation via interference,” Phys. Rev. Lett. 72(5), 629–632 (1994).
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P. G. Mattle, K. Weinfurter, H. Zeilinger, A. Sergienko, A. V. Shih, and Y. Kwiat, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995).
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G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Tunable spatial decoherers for polarization-entangled photons,” Opt. Lett. 31(13), 2057–2059 (2006).
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G. Puentes, G. Waldherr, P. Neumann, G. Balasubramanian, and J. Wrachtrup, “Efficient route to high-bandwidth nanoscale magnetometry using single spins in diamond,” Sci. Rep. 4(1), 4677 (2015).
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A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
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S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
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R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, “Bell-Type Test of Energy-Time Entangled Qutrits,” Phys. Rev. Lett. 93(1), 010503 (2004).
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R. Thew, S. Tanzilli W. Tittel, H. Zbinden, and N. Gisin, “Experimental investigation of the robustness of partially entangled qubits over 11 km,” Phys. Rev. A 66(6), 062304 (2002).
[Crossref]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed Energy-Time Entangled Twin-Photon Source for Quantum Communication,” Phys. Rev. Lett. 82(12), 2594–2597 (1999).
[Crossref]

Zeilinger, A.

S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, “Discrete Tunable Color Entanglement,” Phys. Rev. Lett. 103(25), 253601 (2009).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental Two-Photon, Three-Dimensional Entanglement for Quantum Communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref]

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of Local Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubits,” Phys. Rev. Lett. 85(21), 4418–4421 (2000).
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[Crossref]

Zeilinger, H.

P. G. Mattle, K. Weinfurter, H. Zeilinger, A. Sergienko, A. V. Shih, and Y. Kwiat, “New High-Intensity Source of Polarization-Entangled Photon Pairs,” Phys. Rev. Lett. 75(24), 4337–4341 (1995).
[Crossref]

Zukowski, M.

T. Durt, N. J. Cerf, N. Gisin, and M. Zukowski, “Security of quantum key distribution with entangled qutrits,” Phys. Rev. A 67(1), 012311 (2003).
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D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of Local Realism by Two Entangled N-Dimensional Systems Are Stronger than for Two Qubits,” Phys. Rev. Lett. 85(21), 4418–4421 (2000).
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J. Opt. (1)

G. Sorelli, V. N Shatokhin, and A. Buchleitner, “Universal entanglement loss induced by angular uncertainty,” J. Opt. 22(2), 024002 (2020).
[Crossref]

J. Phys. B: At., Mol. Opt. Phys. (1)

S. Moulieras, M. Lewenstein, and G. Puentes, “Entanglement engineering and topological protection by discrete-time quantum walks,” J. Phys. B: At., Mol. Opt. Phys. 46(10), 104005 (2013).
[Crossref]

Laser Phys. (1)

A. Ling, P. Y. Han, A. Lamas-Limares, and C. Kurtsiefer, “Preparation of Bell state with controlled white noise,” Laser Phys. 16(7), 1140–1144 (2006).
[Crossref]

Nat. Phys. (1)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalised Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

New J. Phys. (7)

B. Jack, J. Leach, H. Ritsch, S. M. Barnet, M. J. Padgett, and S. Franke-Arnold, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 11(10), 103024 (2009).
[Crossref]

G. Puentes, A. Datta, A. Feito, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Entanglement quantification from incomplete measurements: applications using photon-number-resolving weak homodyne detectors,” New J. Phys. 12(3), 033042 (2010).
[Crossref]

K. M. R. Audenaert and M. B. Plenio, “When are correlations quantum?—verification and quantification of entanglement by simple measurements,” New J. Phys. 8(11), 266 (2006).
[Crossref]

J. Eisert, F. G. S. L. Brandão, and K. M. R. Audenaert, “Quantitative entanglement witnesses,” New J. Phys. 9(3), 46 (2007).
[Crossref]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[Crossref]

B. Jack, M. Padgett, and S. Franke-Arnold, “Angular Diffraction,” New J. Phys. 10(10), 103013 (2008).
[Crossref]

G. Puentes, G. Colangelo, R. J. Sewell, and M. W. Mitchell, “Planar squeezing by quantum non-demolition measurement in cold atomic ensembles,” New J. Phys. 15(10), 103031 (2013).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (12)

P. Machado, A. A. Matoso, M. R. Barros, L. Neves, and S. Padua, “Engineering quantum correlations for m×n spatially encoded two-photons states,” Phys. Rev. A 99(6), 063839 (2019).
[Crossref]

P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions,” Phys. Rev. A 64(4), 042315 (2001).
[Crossref]

G. Puentes, A. Aiello, D. Voigt, and J. P. Woerdman, “Maximally entangled mixed-state generation via local operations,” Phys. Rev. A 75(3), 032319 (2007).
[Crossref]

M. Bourennane, A. Karlsson, and G. Bjork, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

T. Durt, N. J. Cerf, N. Gisin, and M. Zukowski, “Security of quantum key distribution with entangled qutrits,” Phys. Rev. A 67(1), 012311 (2003).
[Crossref]

T-C Wei, J. B. Altepeter, D. Branning, P. M. Goldbart, D. F. James, E. Jeffrey, P. G. Kwiat, S. Mukhopadhyay, and N. Peters, “Synthesizing arbitrary two-photon mixed states,” Phys. Rev. A 71(3), 032329 (2005).
[Crossref]

A. K. Jha, B. Jack, E. Yao, J. Leach, R. W. Boyd, G. S. Buller, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Fourier relationship between the angle and angular momentum of entangled photons,” Phys. Rev. A 78(4), 043810 (2008).
[Crossref]

S. M. Barnett and D. T. Pegg, “Quantum theory of rotation angles,” Phys. Rev. A 41(7), 3427–3435 (1990).
[Crossref]

K. Jha, M. N. O’Sullivan, K. W. Clifford Chan, and R. W. Boyd, “Temporal coherence and indistinguishability in two-photon interference effects,” Phys. Rev. A 77(2), 021801 (2008).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. (a) Schematic of proposed experimental setup (see text for details). (b) Two-photon multiple-path diagram showing $N^2$ alternative paths using angular masks containing $N$ slits of width $\alpha$ and separation $\beta$ , with $N(\alpha + \beta ) \leq 2 \pi$ . (c) Angular apertures used to create path-entanglement, and diffraction holograms used to analyze the OAM spectrum. Both the angular aperture and the diffraction hologram are programmed using standard Spatial Light Modulators.
Fig. 2.
Fig. 2. Density matrix representation ( $\hat {\rho }$ ) of pure maximally entangled states, in the complete pathway basis $\{|s,n\rangle |i,m \rangle \}$ ( $n=0,\ldots ,N-1; m=0,\ldots ,N-1$ ), with $N=2$ (see text for details). Left column $\mathrm {Re}[{\hat {\rho }}]$ , right column $\mathrm {Im}[{\hat {\rho }}]$ , (a) $N=2$ , $\theta =0$ , (b) $N=2$ , $\theta =0$ , (c) $N=2$ , $\theta =\pi /4$ , (d) $N=2$ , $\theta =\pi /4$ .
Fig. 3.
Fig. 3. Density matrix representation ( $\hat {\rho }$ ) of pure maximally entangled states, in the complete pathway basis $\{|s,n\rangle |i,m \rangle \}$ ( $n=0,\ldots ,N-1; m=0,\ldots ,N-1$ ), with $N=4$ (see text for details). Left column $\mathrm {Re}[{\hat {\rho }}]$ , right column $\mathrm {Im}[{\hat {\rho }}]$ , (a) $N=4$ , $\theta =0$ , (b) $N=4$ , $\theta =0$ , (c) $N=4$ , $\theta =\pi /4$ , (d) $N=4$ , $\theta =\pi /4$ .
Fig. 4.
Fig. 4. Density matrix representation ( $\hat {\rho }$ ) of pure maximally entangled states, in the complete pathway basis $\{|s,n\rangle |i,m \rangle \}$ ( $n=0,\ldots ,N-1; m=0,\ldots ,N-1$ ), with $N=5$ (see text for details). Left column $\mathrm {Re}[{\hat {\rho }}]$ , right column $\mathrm {Im}[{\hat {\rho }}]$ , (a) $N=5$ , $\theta =0$ , (b) $N=5$ , $\theta =0$ , (c) $N=5$ , $\theta =\pi /4$ , (d) $N=5$ , $\theta =\pi /4$ .
Fig. 5.
Fig. 5. Density matrix ( $\hat {\rho }$ ) representation of pure maximally entangled states in the complete pathway basis $\{|s,n\rangle |i,m \rangle \}$ ( $n=0,\ldots ,N-1; m=0,\ldots ,N-1$ ), with $N=10$ (see text for details). Left column $\mathrm {Re}[{\hat {\rho }}]$ , right column $\mathrm {Im}[{\hat {\rho }}]$ , (a) $N=10$ , $\theta =0$ , (b) $N=10$ , $\theta =0$ , (c) $N=10$ , $\theta =\pi /4$ , (d) $N=10$ , $\theta =\pi /4$ .
Fig. 6.
Fig. 6. Simulated interference fringes, given by Coincidence Count Rates ( $R_{si}$ ) in Eq. (7), for off diagonal elements $\rho _{nm}=\frac {1}{N}e^{i \theta }$ , $\alpha =\pi /10$ , $\beta =\pi /4$ , and $N=6$ angular slits. (a) $l_{i}=2$ , (b) $l_{i}=-2$ , Due to OAM correlations between twin photons the interference pattern has a maximum for $l_{s}=-l_{i}$ . Figures (c)-(f) correspond to different angular separation $\beta$ , for $l_{i}=0$ and $\alpha =\pi /10$ . (c) $\beta =\pi /4$ , (d) $\beta =\pi /7$ , (e) $\beta =\pi /11$ , (f) $\beta =\pi /14$ . As expected the period of the interference pattern decreases as $\beta$ increases (see text for details).
Fig. 7.
Fig. 7. Simulated interference fringes, given by Coincidence Count Rates ( $R_{si}$ ) in Eq. (7), for a reported visibility $V=0.875$ [23], and $N=2$ angular slits. (a) $l_{i}=2$ , (b) $l_{i}=-2$ . Due to OAM correlations between twin photons the interference pattern has a maximum for $l_{s}=-l_{i}$ . Figures (c)-(f) correspond to different angular separation $\beta$ for $l_{i}=0$ and $\alpha =\pi /10$ . (c) $\beta =\pi /6$ , (d) $\beta =\pi /4$ , (e) $\beta =\pi /2$ , (f) $\beta =\pi$ . As expected the period of the interference pattern decreases as $\beta$ increases (see Eq. (7) for details). Our numerical results fully reproduce the experimental results reported in Ref. [23].
Fig. 8.
Fig. 8. Coincidence Count Rate $(R_{s,i})$ given by Eq. (13) for pure non-maximally entangled states produced by imperfect phase-matching in combination with asymmetric slit configuration [24], as a function of $l_{s}$ , for $l_{i}=0$ , $\alpha =\pi /10$ , off diagonal elements $\rho _{nm}=\frac {1}{\sqrt {NM}}e^{i \theta }$ , $N=6$ and $M=3$ angular slits. We consider different slit separations (a) $\beta =\pi /4$ , (b) $\beta =\pi /7$ . Such interference effects are a signature of non-maximal path entanglement in a $D$ -dimensional space spanned by the different path alternatives of dimension $D=N \times M=18$ . As expected the period of the interference pattern decreases as $\beta$ increases (see text for details).

Equations (24)

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| ψ s l = l = + a l | l s | l i ,
A j , n ( ϕ j ) = 1 i f n β α / 2 ϕ j n β + α / 2 e l s e 0 ,
| s , 0 { | i , 0 , | i , 1 , , | i , N 1 } ; | s , 1 { | i , 0 , | i , 1 , , | i , N 1 } ; . | s , N 1 { | i , 0 , | i , 1 , , | i , N 1 } .
ρ ^ = n = 0 N 1 m = 0 N 1 ρ n m | s , n | i , n s , m | i , m | ,
| s , n | i , n = C l c l l 1 2 π π π d ϕ s A s , n ( ϕ s ) e i ( l l ) ϕ s | l × l 1 2 π π π d ϕ i A i , n ( ϕ i ) e i ( l + l ) ϕ i | l ,
A ~ j , n = 1 2 π π π d ϕ A j , n ( ϕ ) e i l j ϕ = α e i l j β n 2 π s i n c ( α 2 l j ) ,
R s i = C 2 α 2 16 π 4 | l = L l = L c l s i n c ( ( l s l ) α / 2 ) s i n c ( ( l i + l ) α / 2 ) | 2 × n = 0 N 1 m = 0 N 1 ρ n m e i β ( l s + l i ) ( n m )
R s i = C 2 α 2 16 π 4 | l = L l = L c l s i n c ( ( l s l ) α / 2 ) s i n c ( ( l i + l ) α / 2 ) | 2 × [ ρ 00 + ρ 11 + 2 ρ 00 ρ 11 μ cos ( β ( l s + l i ) + θ ) ] ,
n = 0 N 1 m = 0 N 1 ρ n m e i β ( l s + l i ) ( n m ) .
V = 2 ρ 00 ρ 11 μ ,
| s , n | i , m = C l c l l 1 2 π π π d ϕ s A s , n ( ϕ s ) e i ( l l ) ϕ s | l × l 1 2 π π π d ϕ i A i , m ( ϕ i ) e i ( l + l ) ϕ i | l .
ρ ^ = n , n = 0 N 1 m , m = 0 M 1 ρ n m , n m | s , n | i , m s , n | i , m | ,
R s i = C 2 α 2 16 π 4 | l = L l = L c l s i n c ( ( l s l ) α / 2 ) s i n c ( ( l i + l ) α / 2 ) | 2 × n , n = 0 N 1 m , m = 0 M 1 ρ n m , n m e i β l s ( n n ) e i β l i ( m m )
C ( ψ A B ) = 2 ν D 1 ν D 2 [ 1 T r ( ρ A ^ 2 ) ] ,
E min = min ρ ^ { E ( ρ ^ ) : T r ( ρ ^ M i ) = m i } ,
ρ ^ T A 1 = max H = 1 T r ( H ρ ^ T A ) = max H = 1 T r ( H T A ρ ^ ) ,
N min = log min ρ ^ { max H { T r ( H T A ρ ^ ) | H = 1 } : T r ( ρ ^ M i ) = m i } .
N min = log max H { min ρ ^ { T r ( H T A ρ ^ ) : T r ( ρ ^ M i ) = m i } : H = 1 } .
H T A i ν i M i .
T r ( H T A ρ ^ ) i ν i T r ( M i ρ ^ ) = i ν i m i .
N min log max H { × max ν i { i ν i m i : H T A i ν i M i } : H = 1 } .
m a x i m i z e log ( i ν i m i ) , s u b j e c t t o H T A i ν i M i , a n d I H I ,
M n = Π j s Π k i .
Π j s , i = | l j l j | s , i ,

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