Abstract

Unitary transformations are the fundamental building blocks of gates and operations in quantum information processing, allowing the complete manipulation of quantum systems in a coherent manner. In the case of photons, optical elements that can perform unitary transformations are readily available only for some degrees of freedom, e.g., wave plates for polarization. However, for high-dimensional states encoded in the transverse spatial modes of light, performing arbitrary unitary transformations remains a challenging task for both theoretical proposals and actual implementations. Following the idea of multi-plane light conversion, we show that it is possible to perform a broad variety of unitary operations at high quality by using only a few phase modulation planes. More importantly, we experimentally implement several high-dimensional quantum gates for up to five-dimensional states encoded in the full-field mode structure of photons. In particular, we realize cyclic and quantum Fourier transformations, known as Pauli $ \hat X $-gates and Hadamard $ \hat H $-gates, respectively, with an average visibility of more than 90%. In addition, we demonstrate near-perfect “unitarity” by means of quantum process tomography, unveiling a process purity of 99%. Last, we demonstrate the benefit of the two independent spatial degrees of freedom, i.e., azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Thus, our demonstrations open up new paths to implement high-dimensional quantum operations, which can be applied to various tasks in quantum communication, computation, and sensing schemes.

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Near-perfect measuring of full-field transverse-spatial modes of light

Markus Hiekkamäki, Shashi Prabhakar, and Robert Fickler
Opt. Express 27(22) 31456-31464 (2019)

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2020 (1)

R. Fickler, F. Bouchard, E. Giese, V. Grillo, G. Leuchs, and E. Karimi, “Full-field mode sorter using two optimized phase transformations for high-dimensional quantum cryptography,”J. Opt. 22, 024001 (2020).
[Crossref]

2019 (5)

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

X. Gao, M. Krenn, J. Kysela, and A. Zeilinger, “Arbitrary d -dimensional Pauli X gates of a flying qudit,” Phys. Rev. A 99, 023825 (2019).
[Crossref]

P. Imany, J. A. Jaramillo-Villegas, M. S. Alshaykh, J. M. Lukens, O. D. Odele, A. J. Moore, D. E. Leaird, M. Qi, and A. M. Weiner, “High-dimensional optical quantum logic in large operational spaces,” Quantum Inf. 5, 1–10 (2019).
[Crossref]

F. Bouchard, F. Hufnagel, D. Koutný, A. Abbas, A. Sit, K. Heshami, R. Fickler, and E. Karimi, “Quantum process tomography of a high-dimensional quantum communication channel,” Quantum 3, 138 (2019).
[Crossref]

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Overcoming noise in entanglement distribution,” Phys. Rev. X 9, 041042 (2019).
[Crossref]

2018 (5)

J. Bavaresco, N. Herrera Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, “Measurements in two bases are sufficient for certifying high-dimensional entanglement,” Nat. Phys. 14, 1032–1037 (2018).
[Crossref]

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

X. Gu, M. Krenn, M. Erhard, and A. Zeilinger, “Gouy phase radial mode sorter for light: concepts and experiments,” Phys. Rev. Lett. 120, 103601 (2018).
[Crossref]

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018).
[Crossref]

2017 (9)

Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. Hashemi Rafsanjani, A. E. Willner, and R. W. Boyd, “Sorting photons by radial quantum number,” Phys. Rev. Lett. 119, 263602 (2017).
[Crossref]

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-dimensional single-photon quantum gates: concepts and experiments,” Phys. Rev. Lett. 119, 180510 (2017).
[Crossref]

P. Boucher, N. Barré, O. Pinel, G. Labroille, and N. Treps, “Continuous axial scanning of a Gaussian beam via beam steering,” Opt. Express 25, 23060–23069 (2017).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, and C. Marquardt, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006–1010 (2017).
[Crossref]

F. Bouchard, R. Fickler, R. W. Boyd, and E. Karimi, “High-dimensional quantum cloning and applications to quantum hacking,” Sci. Adv. 3, e1601915 (2017).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

2016 (4)

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10, 010,” Proc. Natl. Acad. Sci. USA 113, 13642–13647 (2016).
[Crossref]

F. Schlederer, M. Krenn, R. Fickler, M. Malik, and A. Zeilinger, “Cyclic transformation of orbital angular momentum modes,” New J. Phys. 18, 043019 (2016).
[Crossref]

2015 (7)

C. Schaeff, R. Polster, M. Huber, S. Ramelow, and A. Zeilinger, “Experimental access to higher-dimensional entangled quantum systems using integrated optics,” Optica 2, 523–529 (2015).
[Crossref]

W. N. Plick and M. Krenn, “Physical meaning of the radial index of Laguerre-Gauss beams,” Phys. Rev. A 92, 063841 (2015).
[Crossref]

M. Giustina, S. Versteegh, M. A. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of Bell’s theorem with entangled photons,” Phys. Rev. Lett. 115, 250401 (2015).
[Crossref]

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F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko, D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd, and L. Marrucci, “Quantum walks and wavepacket dynamics on a lattice with twisted photons,” Sci. Adv. 1, e1500087 (2015).
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M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
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M. G. McLaren, F. S. Roux, and A. Forbes, “Realising high-dimensional quantum entanglement with orbital angular momentum,” South African J. Sci. 111, 1–9 (2015).
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2014 (4)

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
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E. Karimi, R. W. Boyd, P. De La Hoz, H. De Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
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E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
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G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J.-F. Morizur, “Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion,” Opt. Express 22, 15599–15607 (2014).
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2013 (3)

2012 (2)

2010 (2)

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
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J.-F. Morizur, L. Nicholls, P. Jian, S. Armstrong, N. Treps, B. Hage, M. Hsu, W. Bowen, J. Janousek, and H.-A. Bachor, “Programmable unitary spatial mode manipulation,” J. Opt. Soc. Am. A 27, 2524–2531 (2010).
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2009 (3)

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
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S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Universal unitary gate for single-photon spin-orbit four-dimensional states,” Phys. Rev. A 80, 022326 (2009).
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B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
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2007 (1)

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
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2005 (1)

A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).
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2003 (1)

J. P. Dowling and G. J. Milburn, “Quantum technology: the second quantum revolution,” Philos. Trans. R. Soc. London, Ser. A 361, 1655–1674 (2003).
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2002 (3)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
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N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
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2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
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2000 (1)

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
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1999 (1)

J. Courtial and M. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre-Gaussian laser modes,” Opt. Commun. 159, 13–18 (1999).
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G. Turnbull, D. Robertson, G. Smith, L. Allen, and M. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
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N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
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Abbas, A.

F. Bouchard, F. Hufnagel, D. Koutný, A. Abbas, A. Sit, K. Heshami, R. Fickler, and E. Karimi, “Quantum process tomography of a high-dimensional quantum communication channel,” Quantum 3, 138 (2019).
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Abellán, C.

M. Giustina, S. Versteegh, M. A. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of Bell’s theorem with entangled photons,” Phys. Rev. Lett. 115, 250401 (2015).
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E. Karimi, R. W. Boyd, P. De La Hoz, H. De Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
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G. Turnbull, D. Robertson, G. Smith, L. Allen, and M. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127, 183–188 (1996).
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L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, “Strong loophole-free test of local realism,” Phys. Rev. Lett. 115, 250402 (2015).
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B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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M. Giustina, S. Versteegh, M. A. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of Bell’s theorem with entangled photons,” Phys. Rev. Lett. 115, 250401 (2015).
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Baker, M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’brien, A. Gilchrist, and A. G. White, “Simplifying quantum logic using higher-dimensional Hilbert spaces,” Nat. Phys. 5, 134–140 (2009).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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J. Bavaresco, N. Herrera Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, and M. Huber, “Measurements in two bases are sufficient for certifying high-dimensional entanglement,” Nat. Phys. 14, 1032–1037 (2018).
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G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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Bent, N.

E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, “Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference,” Phys. Rev. A 89, 013829 (2014).
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E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
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G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
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Berry, M. V.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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Beyer, J.

M. Giustina, S. Versteegh, M. A. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-Å. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, “Significant-loophole-free test of Bell’s theorem with entangled photons,” Phys. Rev. Lett. 115, 250401 (2015).
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L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, “Strong loophole-free test of local realism,” Phys. Rev. Lett. 115, 250402 (2015).
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L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, “Strong loophole-free test of local realism,” Phys. Rev. Lett. 115, 250402 (2015).
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H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
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Bolduc, E.

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Supplementary Material (1)

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» Supplement 1       theory and additional measurements

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Figures (6)

Fig. 1.
Fig. 1. Presentation of the multi-plane light-conversion (MPLC) technique. (a) Simulated evolution of the input modes at different stages of the MPLC setup is shown for the case of three phase elements. We note that the input modes are all co-propagating and therefore experience the same three phase manipulations. (b) Simulated visibility, transformation efficiency, and process purity of the cyclic transformation are shown as a function of the number of phase elements for different numbers of OAM modes.
Fig. 2.
Fig. 2. Experimental implementation of the three-dimensional cyclic transformation. (a) Simplified experimental setup showing the implementation of the three-dimensional $ \hat X $-gate using three phase elements. Photons from the source are spatially filtered to the fundamental Gaussian mode using a single-mode fiber (black wire). The beam is then made incident on SLM-A, where both the phase and amplitude are manipulated in order to generate the appropriate LG mode. These modes are then fed into the MPLC system consisting of SLM-B, displaying three phase patterns, and a mirror (M). The beam bounces off between the SLM and the mirror three times before it exits the system and is detected using SLM-C with the intensity-flattening technique introduced only recently in Ref. [43]. (b) Theoretical prediction of a tomographically complete set of measurements for the theoretical three-dimensional $ \hat X $-gate. The input and output states are chosen from the three-dimensional MUBs labeled here by I, II, III, and IV. (c) Experimental correlation matrix for different MUBs obtained from the MPLC system described in (a).
Fig. 3.
Fig. 3. Quantum process tomography of the three-dimensional cyclic transformation. The real (a) and imaginary (b) parts of the theoretical process matrix, $ {\chi _{{\rm th}}} $, for the three-dimensional $ \hat X $-gate are shown in the Gell–Mann matrix basis. The real (c) and imaginary (d) parts of the experimentally reconstructed process matrix, $ {\chi _{{\rm exp}}} $, for the experimental implementation of the three-dimensional $ \hat X $-gate using three phase elements and OAM modes $ \ell = - 1,0,1 $ are shown in the Gell–Mann matrix basis.
Fig. 4.
Fig. 4. Experimental results for the OAM transformations using a single SLM for generation, manipulation (eight phase planes), and detection. Crosstalk matrices are shown for a three-dimensional Hadamard gate ($ {\hat H_1} $), converting input beams from the computational basis to another mutually unbiased basis (here, the angle basis) (a) and a Pauli $ {\hat X^1} $-gate in dimension five (c). We further realize combined $ \hat X\hat H $ operations performing a cyclic operation on a set of input modes that is transformed to another MUB in dimension three ($ {\hat X^2}{\hat H_2} $-gate) (b) and dimension five ($ {\hat X^1}{\hat H_2} $-gate) (d). Note that the quality for measurements of complex superposition modes in large Hilbert spaces as in (d) decreases due to limited resolution of the holograms for the single-SLM implementation.
Fig. 5.
Fig. 5. Experimental results of different $p$-mode transformations, again using only one SLM for generation, manipulation, and detection. A cyclic transformation $ {\hat X^1} $ on $ p $-only modes in the computational basis is shown for dimension two in (a) and dimension three in (b). In addition, we implement also combined $ \hat X \hat H $-gates in dimension three, in particular, a $ {\hat X^1}{\hat H_2} $-gate shown in (c) and a $ {\hat X^1}\hat H_1^\dagger $-gate shown in (d). Note that measuring $p$-modes properly in different mutually unbiased bases has become possible only recently [43].
Fig. 6.
Fig. 6. Single-photon CNOT gate using the radial modes $ |p\rangle $ as a control qubit for the manipulation of the azimuthal modes $ |\ell \rangle $. Experimental crosstalk matrix with input and output modes labeled by the states $ |p,\ell \rangle $. A visibility of 94.6% is achieved for the single-photon CNOT gate with a MPLC system consisting of a generation SLM, an SLM used for the manipulation (consisting of three phase planes), and one SLM as the detection SLM. The quantum circuit diagram of a CNOT gate as used frequently in quantum information science is also shown in the top right corner.

Equations (8)

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ρ f = U ^ ρ i U ^ .
Δ Φ t ( x , y ) = arg ( r = s o rst ( x , y ) e i ϕ rst ) ,
X ^ m | l = | ( l + m ) m o d ( d ) ,
H ^ m | Ψ l = 1 d k = 0 d 1 ω d ( l k + ( m 1 ) k 2 ) | k ,
c X ^ ( | p | l ) = ( I ^ X ^ p ) | p | l = | p | ( l + p ) m o d ( d ) .
V = i C ii / ij C ij ,
E ( ρ ) = i , j χ i , j σ ^ i ρ σ ^ j ,
ρ E = ( I ^ E ) | Ψ Ψ | ,