Abstract

A tomographic measurement is a ubiquitous tool for estimating the properties of quantum states, and its application is known as quantum state tomography (QST). The process involves manipulating single photons in a sequence of projective measurements, often to construct a density matrix from which other information can be inferred, and is as laborious as it is complex. Here we unravel the steps of a QST and outline how it may be demonstrated in a fast and simple manner with intense (classical) light. We use scalar beams in a time reversal approach to simulate the outcome of a QST and exploit non-separability in classical vector beams as a means to treat the latter as a “classically entangled” state for illustrating QSTs directly. We provide a complete do-it-yourself resource for the practical implementation of this approach, complete with tutorial video, which we hope will facilitate the introduction of this core quantum tool into teaching and research laboratories alike. Our work highlights the value of using intense classical light as a means to study quantum systems and in the process provides a tutorial on the fundamentals of QSTs.

© 2019 Optical Society of America

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2018 (15)

J. Bavaresco, N. H. 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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J. G. Titchener, M. Gräfe, R. Heilmann, A. S. Solntsev, A. Szameit, and A. A. Sukhorukov, “Scalable on-chip quantum state tomography,” npj Quantum Inf. 4, 19 (2018).
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K. Wang, J. G. Titchener, S. S. Kruk, L. Xu, H.-P. Chung, M. Parry, I. I. Kravchenko, Y.-H. Chen, A. S. Solntsev, Y. S. Kivshar, D. N. Neshev, and A. A. Sukhorukov, “Quantum metasurface for multi-photon interference and state reconstruction,” Science 361, 1104–1108 (2018).
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M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

M. Arruda, W. Soares, S. Walborn, D. Tasca, A. Kanaan, R. M. de Araújo, and P. Ribeiro, “Klyshko’s advanced-wave picture in stimulated parametric down-conversion with a spatially structured pump beam,” Phys. Rev. A 98, 023850 (2018).
[Crossref]

P.-A. Moreau, E. Toninelli, T. Gregory, and M. J. Padgett, “Ghost imaging using optical correlations,” Laser Photon. Rev. 12, 1700143 (2018).
[Crossref]

M. G. Nassiri and E. Brasselet, “Multispectral management of the photon orbital angular momentum,” Phys. Rev. Lett. 121, 213901 (2018).
[Crossref]

E. Brasselet, “Tunable high-resolution macroscopic self-engineered geometric phase optical elements,” Phys. Rev. Lett. 121, 033901 (2018).
[Crossref]

M. Rafayelyan and E. Brasselet, “Spin-to-orbital angular momentum mapping of polychromatic light,” Phys. Rev. Lett. 120, 213903 (2018).
[Crossref]

A. Ambrosio, “Structuring visible light with dielectric metasurfaces,” J. Opt. 20, 113002 (2018).
[Crossref]

H. Sroor, N. Lisa, D. Naidoo, I. Litvin, and A. Forbes, “Purity of vector vortex beams through a birefringent amplifier,” Phys. Rev. Appl. 9, 044010(2018).
[Crossref]

C. Rosales-Guzmán, B. Ndagano, and A. Forbes, “A review of complex vector light fields and their applications,” J. Opt. 20, 123001 (2018).
[Crossref]

M. Delmans and J. Haseloff, “μCube: a framework for 3D printable optomechanics,” J. Open Hardware 2, 2 (2018).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2018).
[Crossref]

P. Li, S. Zhang, and X. Zhang, “Classically high-dimensional correlation: simulation of high-dimensional entanglement,” Opt. Express 26, 31413–31429 (2018).
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2017 (17)

R. C. Devlin, A. Ambrosio, D. Wintz, S. L. Oscurato, A. Y. Zhu, M. Khorasaninejad, J. Oh, P. Maddalena, and F. Capasso, “Spin-to-orbital angular momentum conversion in dielectric metasurfaces,” Opt. Express 25, 377–393 (2017).
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A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Gunthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006–1010 (2017).
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C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25, 25697–25706 (2017).
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L. J. Salazar-Serrano, J. P. Torres, and A. Valencia, “A 3D printed toolbox for opto-mechanical components,” PloS One 12, e0169832 (2017).
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B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
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I. Nape, B. Ndagano, and A. Forbes, “Erasing the orbital angular momentum information of a photon,” Phys. Rev. A 95, 053859 (2017).
[Crossref]

M. McLaren and A. Forbes, “Digital spiral-slit for bi-photon imaging,” J. Opt. 19, 044006 (2017).
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F. Bouchard, R. Fickler, R. W. Boyd, and E. Karimi, “High-dimensional quantum cloning and applications to quantum hacking,” Sci. Adv. 3, e1601915 (2017).
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Y. Zhang, M. Agnew, T. Roger, F. S. Roux, T. Konrad, D. Faccio, J. Leach, and A. Forbes, “Simultaneous entanglement swapping of multiple orbital angular momentum states of light,” Nat. Commun. 8, 632 (2017).
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R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to-orbital angular momentum conversion of light,” Science 358, 896–901 (2017).
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K. Subramanian and N. K. Viswanathan, “Measuring correlations in non-separable vector beams using projective measurements,” Opt. Commun. 399, 45–51 (2017).
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B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. 13, 397–402 (2017).
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X.-F. Qian, A. N. Vamivakas, and J. H. Eberly, “Emerging connections: classical and quantum optics,” Opt. Photon. News 28(10), 34–41 (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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A. E. Willner, Y. Ren, G. Xie, Y. Yan, L. Li, Z. Zhao, J. Wang, M. Tur, A. F. Molisch, and S. Ashrafi, “Recent advances in high-capacity free-space optical and radio-frequency communications using orbital angular momentum multiplexing,” Philos. Trans. R. Soc. London Ser. A 375, 20150439 (2017).
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M. Krenn, M. Malik, M. Erhard, and A. Zeilinger, “Orbital angular momentum of photons and the entanglement of Laguerre-Gaussian modes,” Philos. Trans. R. Soc. London Ser. A 375, 20150442 (2017).
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H. Sosa-Martinez, N. Lysne, C. Baldwin, A. Kalev, I. Deutsch, and P. Jessen, “Experimental study of optimal measurements for quantum state tomography,” Phys. Rev. Lett. 119, 150401 (2017).
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2016 (14)

S. S. Straupe, “Adaptive quantum tomography,” JETP Lett. 104, 510–522 (2016).
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M. Ghalaii, M. Afsary, S. Alipour, and A. Rezakhani, “Quantum imaging as an ancilla-assisted process tomography,” Phys. Rev. A 94, 042102 (2016).
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Ö. Bayraktar, M. Swillo, C. Canalias, and G. Björk, “Quantum-polarization state tomography,” Phys. Rev. A 94, 020105 (2016).
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Y. Zhang, S. Prabhakar, C. Rosales-Guzmán, F. S. Roux, E. Karimi, and A. Forbes, “Hong-Ou-Mandel interference of entangled Hermite-Gauss modes,” Phys. Rev. A 94, 033855 (2016).
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Y. Zhang, F. S. Roux, T. Konrad, M. Agnew, J. Leach, and A. Forbes, “Engineering two-photon high-dimensional states through quantum interference,” Sci. Adv. 2, e1501165 (2016).
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C. Samlan and N. K. Viswanathan, “Generation of vector beams using a double-wedge depolarizer: non-quantum entanglement,” Opt. Lasers Eng. 82, 135–140 (2016).
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R. Schmied, “Quantum state tomography of a single qubit: comparison of methods,” J. Mod. Opt. 63, 1744–1758 (2016).
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M. J. Escuti, J. Kim, and M. W. Kudenov, “Controlling light with geometric-phase holograms,” Opt. Photon. News 27(2), 22–29 (2016).
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R. C. Devlin, M. Khorasaninejad, W. T. Chen, J. Oh, and F. Capasso, “Broadband high-efficiency dielectric metasurfaces for the visible spectrum,” Proc. Natl. Acad. Sci. USA 113, 10473–10478 (2016).
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D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
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J. P. Sharkey, D. C. W. Foo, A. Kabla, J. J. Baumberg, and R. W. Bowman, “A one-piece 3D printed flexure translation stage for open-source microscopy,” Rev. Sci. Instrum. 87, 025104 (2016).
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A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
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P. Li, B. Wang, and X. Zhang, “High-dimensional encoding based on classical nonseparability,” Opt. Express 24, 15143–15159 (2016).
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B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
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2015 (19)

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
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G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, “4 × 20  Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de) multiplexer,” Opt. Lett. 40, 1980–1983 (2015).
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B. Ndagano, R. Brüning, M. McLaren, M. Duparré, and A. Forbes, “Fiber propagation of vector modes,” Opt. Express 23, 17330–17336 (2015).
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X.-F. Qian, B. Little, J. C. Howell, and J. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
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S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, “Classically entangled optical beams for high-speed kinematic sensing,” Optica 2, 864–868 (2015).
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D. J. Lum, S. H. Knarr, and J. C. Howell, “Fast Hadamard transforms for compressive sensing of joint systems: measurement of a 3.2 million-dimensional bi-photon probability distribution,” Opt. Express 23, 27636–27649 (2015).
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G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. 40, 4887–4890 (2015).
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M. A. Hossain, J. Canning, K. Cook, and A. Jamalipour, “Smartphone laser beam spatial profiler,” Opt. Lett. 40, 5156–5159 (2015).
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A. P. Zwicker, J. Bloom, R. Albertson, and S. Gershman, “The suitability of 3D printed plastic parts for laboratory use,” Am. J. Phys. 83, 281–285(2015).
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K. Y. Bliokh, F. J. Rodrguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
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F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
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X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, S. Wen, and D. Fan, “Hybrid-order Poincaré sphere,” Phys. Rev. A 91, 023801 (2015).
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G. Milione, A. Dudley, T. A. Nguyen, O. Chakraborty, E. Karimi, A. Forbes, and R. R. Alfano, “Measuring the self-healing of the spatially inhomogeneous states of polarization of vector Bessel beams,” J. Opt. 17, 035617 (2015).
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M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
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A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10, 937–943 (2015).
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E. Karimi and R. W. Boyd, “Classical entanglement?” Science 350, 1172–1173 (2015).
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A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
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N. Bent, H. Qassim, A. Tahir, D. Sych, G. Leuchs, L. Sánchez-Soto, E. Karimi, and R. Boyd, “Experimental realization of quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (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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2014 (14)

H. Zhu, “Quantum state estimation with informationally overcomplete measurements,” Phys. Rev. A 90, 012115 (2014).
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E. Karimi, R. 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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P. Ghose and A. Mukherjee, “Entanglement in classical optics,” Rev. Theor. Sci. 2, 274–288 (2014).
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F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
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R. S. Aspden, D. S. Tasca, A. Forbes, R. W. Boyd, and M. J. Padgett, “Experimental demonstration of Klyshko’s advanced-wave picture using a coincidence-count based, camera-enabled imaging system,” J. Mod. Opt. 61, 547–551 (2014).
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L. J. Pereira, A. Z. Khoury, and K. Dechoum, “Quantum and classical separability of spin-orbit laser modes,” Phys. Rev. A 90, 053842 (2014).
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D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345, 298–302 (2014).
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N. F. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
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M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248(2014).
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A. Vallés, V. D’Ambrosio, M. Hendrych, M. Mičuda, L. Marrucci, F. Sciarrino, and J. P. Torres, “Generation of tunable entanglement and violation of a Bell-like inequality between different degrees of freedom of a single photon,” Phys. Rev. A 90, 052326 (2014).
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G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
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R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Quantum entanglement of complex photon polarization patterns in vector beams,” Phys. Rev. A 89, 060301 (2014).
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G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
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Y. Zhang, M. McLaren, F. S. Roux, and A. Forbes, “Simulating quantum state engineering in spontaneous parametric down-conversion using classical light,” Opt. Express 22, 17039–17049 (2014).
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2013 (11)

D. Flamm, C. Schulze, D. Naidoo, S. Schroter, A. Forbes, and M. Duparre, “All-digital holographic tool for mode excitation and analysis in optical fibers,” J. Lightwave Technol. 31, 1023–1032 (2013).
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A. Dudley, Y. Li, T. Mhlanga, M. Escuti, and A. Forbes, “Generating and measuring nondiffracting vector Bessel beams,” Opt. Lett. 38, 3429–3432 (2013).
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C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
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N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340, 724–726 (2013).
[Crossref]

M. McLaren, J. Romero, M. J. Padgett, F. S. Roux, and A. Forbes, “Two-photon optics of Bessel-Gaussian modes,” Phys. Rev. A 88, 033818 (2013).
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D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
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D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
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K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
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M. Krenn, R. Fickler, M. Huber, R. Lapkiewicz, W. Plick, S. Ramelow, and A. Zeilinger, “Entangled singularity patterns of photons in Ince-Gauss modes,” Phys. Rev. A 87, 012326 (2013).
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M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lütkenhaus, and A. Forbes, “Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,” Phys. Rev. A 88, 032305 (2013).
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K. Banaszek, M. Cramer, and D. Gross, “Focus on quantum tomography,” New J. Phys. 15, 125020 (2013).
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2012 (10)

M. Hendrych, R. Gallego, M. Mičuda, N. Brunner, A. Acín, and J. P. Torres, “Experimental estimation of the dimension of classical and quantum systems,” Nat. Phys. 8, 588–591 (2012).
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J. Ahrens, P. Badziag, A. Cabello, and M. Bourennane, “Experimental device-independent tests of classical and quantum dimensions,” Nat. Phys. 8, 592–595 (2012).
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V. Salakhutdinov, E. Eliel, and W. Löffler, “Full-field quantum correlations of spatially entangled photons,” Phys. Rev. Lett. 108, 173604 (2012).
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G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
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F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936(2012).
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V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961 (2012).
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F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012).
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D. Flamm, D. Naidoo, C. Schulze, A. Forbes, and M. Duparré, “Mode analysis with a spatial light modulator as a correlation filter,” Opt. Lett. 37, 2478–2480 (2012).
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M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20, 23589–23597 (2012).
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C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20, 27866–27873 (2012).
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2011 (11)

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001(2011).
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R. Keil, F. Dreisow, M. Heinrich, A. Tünnermann, S. Nolte, and A. Szameit, “Classical characterization of biphoton correlation in waveguide lattices,” Phys. Rev. A 83, 013808 (2011).
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A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19, 9714–9736 (2011).
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A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
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X.-F. Qian and J. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
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N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
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G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
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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 generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
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M. Agnew, J. Leach, M. McLaren, F. S. Roux, and R. W. Boyd, “Tomography of the quantum state of photons entangled in high dimensions,” Phys. Rev. A 84, 062101 (2011).
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T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
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F. Mallet, M. Castellanos-Beltran, H. Ku, S. Glancy, E. Knill, K. Irwin, G. Hilton, L. Vale, and K. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011).
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2010 (7)

D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, “Quantum state tomography via compressed sensing,” Phys. Rev. Lett. 105, 150401 (2010).
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C. Borges, M. Hor-Meyll, J. Huguenin, and A. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
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B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
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E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
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M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nat. Commun. 1, 149 (2010).
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R. Keil, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. Tünnermann, “Photon correlations in two-dimensional waveguide arrays and their classical estimate,” Phys. Rev. A 81, 023834 (2010).
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E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81, 052317 (2010).
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2009 (13)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
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J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. Boyd, A. Jha, S. Barnett, S. Franke-Arnold, and M. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17, 8287–8293 (2009).
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E. Nagali, F. Sciarrino, F. De Martini, B. Piccirillo, E. Karimi, L. Marrucci, and E. Santamato, “Polarization control of single photon quantum orbital angular momentum states,” Opt. Express 17, 18745–18759 (2009).
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X.-S. Ma, A. Qarry, J. Kofler, T. Jennewein, and A. Zeilinger, “Experimental violation of a Bell inequality with two different degrees of freedom of entangled particle pairs,” Phys. Rev. A 79, 042101 (2009).
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L. Neves, G. Lima, A. Delgado, and C. Saavedra, “Hybrid photonic entanglement: realization, characterization, and applications,” Phys. Rev. A 80, 042322 (2009).
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E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009).
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B. Gadway, E. Galvez, and F. De Zela, “Bell-inequality violations with single photons entangled in momentum and polarization,” J. Phys. B 42, 015503 (2009).
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A. Luis, “Coherence, polarization, and entanglement for classical light fields,” Opt. Commun. 282, 3665–3670 (2009).
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Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. 102, 253904 (2009).
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E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett. 103, 103903 (2009).
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B. Jack, J. Leach, H. Ritsch, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 11, 103024 (2009).
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A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–332 (2009).
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2008 (5)

S. Luo, “Using measurement-induced disturbance to characterize correlations as classical or quantum,” Phys. Rev. A 77, 022301 (2008).
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N. Brunner, S. Pironio, A. Acin, N. Gisin, A. A. Méthot, and V. Scarani, “Testing the dimension of Hilbert spaces,” Phys. Rev. Lett. 100, 210503 (2008).
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M. D. de Burgh, N. K. Langford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,” Phys. Rev. A 78, 052122(2008).
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S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
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R. Meyers, K. S. Deacon, and Y. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801 (2008).
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2007 (2)

L. M. Johansen, “Quantum theory of successive projective measurements,” Phys. Rev. A 76, 012119 (2007).
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C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
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2006 (5)

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Other (10)

Supplemental material to this publication including the video of the system in action, 3D-designs of the roto-flip stages, the Arduino firmware, and the LabVIEW programs, can be found at the following links: https://doi.org/10.6084/m9.figshare.7035506, https://doi.org/10.6084/m9.figshare.7035509, and https://doi.org/10.6084/m9.figshare.7035518.

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E. Otte, I. Nape, C. Rosales-Guzmán, A. Vallés, C. Denz, and A. Forbes, “Recovery of local entanglement in self-healing vector vortex Bessel beams,” arXiv:1805.08179 (2018).

L. Banchi, W. S. Kolthammer, and M. Kim, “Multiphoton tomography with linear optics and photon counting,” arXiv:1806.02436 (2018).

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A. Lvovsky and M. Raymer, “Continuous-variable quantum-state tomography of optical fields and photons,” in Quantum Information with Continuous Variables of Atoms and Light (World Scientific, 2007), pp. 409–433.

A. Chefles, “12 quantum states: discrimination and classical information transmission. A review of experimental progress,” in Quantum State Estimation (Springer, 2004), pp. 467–511.

C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators (SPIE, 2017).

A. Forbes, Laser Beam Propagation: Generation and Propagation of Customized Light (CRC Press, 2014).

Supplementary Material (4)

NameDescription
» Code 1       These programmes were tested using National Instruments LabVIEW 2017 and Vision Sofware 2017. Third party dependencies (provided as part of this code) are also required and can be installed using the National Instruments VI Package Manager.
» Code 2       This Arduino code was tested using the Arduino IDE version 1.8.5.This sketch is the firmware that allows the roto-flip stages to communicate and be controlled by LabVIEW via the serial port.
» Code 3       The 3D-design files and STL files. Further information is provided in the readme file.
» Visualization 1       A video demonstration of our automated tomography system in action is provided.

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Figures (41)

Figure 1.
Figure 1. QST attempts to reconstruct a potentially complex quantum state by a series of simple projective measurements. This is analogous to trying to reconstruct a complicated object by considering only its shadow from various angles.
Figure 2.
Figure 2. Intuitive description of the Poincaré sphere. (a) Control over the amplitude parameter θ allows one to continuously change the position of the qubit (indicated by the arrow) on the unit circle, resulting in a change of the relative amplitudes of left- and right-circular polarizations. (b) Control over the phase parameter ϕ allows one to continuously change the position of the qubit (indicated by the arrow) on a different unit circle, this time resulting in a rotation of the polarization state. This is demonstrated for θ = π / 2 in Eq. (2). (c) Simultaneous control of phase and amplitude results in a description of the general qubit in Eq. (2) on a unit sphere where the poles are the polarization states.
Figure 3.
Figure 3. Elementary rotations on the surface of the Poincaré sphere. The qubit state, indicated by the position vector colored orange, can be rotated on the surface of the sphere about the (a)  x axis, (b)  y axis, and (c)  z axis.
Figure 4.
Figure 4. Graphical representation of the density matrix in terms of its real and imaginary components for the (a) pure state and (b) mixed state examples as given in the main body text.
Figure 5.
Figure 5. Graphical representation of orthogonal and mutually unbiased states used in the QST projections. Here only the polarization matters, shown overlaid on a Gaussian mode.
Figure 6.
Figure 6. (a) Intensities and (b) phase maps of OAM modes carrying, from left to right, = 3 , 2 , , + 2 , and + 3 units of OAM.
Figure 7.
Figure 7. (a) Representation of polarization states on the Poincaré sphere. The poles represent the eigenstates of the basis, from which all other states are constructed. An arbitrary polarization state thus maps to a point on the surface of the Poincaré sphere. (b) The normalized outcomes of projective measurements onto the eigenstates of the Pauli matrices, for given horizontally polarized state. (c) Equivalently, one can construct a similar sphere, a Bloch sphere, where the poles correspond to OAM eigenstates. Here, the eigenstates of the Pauli matrices correspond to spatial modes. (d) From projective measurement onto these spatial modes, one performs the quantum-state tomographic measurement of an OAM state ( | + | ) / 2 with = 1 .
Figure 8.
Figure 8. Bloch sphere description of an OAM qubit in Eq. (23) with 1 = 2 = 1 . Control over the amplitude parameter θ and phase parameter ϕ , as shown in (a) and (b), respectively, leads to (c) a description of the general OAM qubit on a unit sphere where the poles are the pure OAM states.
Figure 9.
Figure 9. Eigenvectors and eigenvalues of the identity and Pauli matrices in the OAM basis.
Figure 10.
Figure 10. Detection of a spatial mode requires a “pattern sensitive” detector. This is achieved by exploiting the reciprocity of light, passing the incoming beam backward through the hologram that would detect it.
Figure 11.
Figure 11. (a) Graphical representation of the outcomes of the projective measurements of a QST on the state | ψ = ( | 1 + | 1 ) / 2 , shown in the inset. (b) Based on these tomographic measurements, one reconstructs the density matrix of the state.
Figure 12.
Figure 12. Graphical representation of the orthogonal and MUBs used in the QST projections for (a) polarization and (b) OAM modes. In the case of the latter, the polarization is no longer important and so is shown as horizontal everywhere. The patterns are shown as intensity functions, while the actual projections are often done with phase-only approximations to these.
Figure 13.
Figure 13. Description of qubit pairs on the higher order Bloch sphere. States on the surface of the higher order Bloch sphere are constructed from the tensor product of qubit states from two Bloch spheres, each describing a subsystem (photon). The entire space is four dimensional, shown here as two-dimensional subspaces for visualization purposes.
Figure 14.
Figure 14. Quantum-state tomographic measurement of a two-photon state. (a) An experimental setup to generate entangled two-photon states through spontaneous parametric downconversion in a nonlinear crystal (NLC). The downconverted photons travel to two OAM analyzers, and their quantum states are measured in coincidence. (b) Shows a two-qubit QST where the projected state of each photon in the entangled pair is indicated by its phase map. The color of each box represents the normalized coincidence counts for a given set of projection on the two-photon state. Using the tomographic data, the density matrix is computed, and its real and imaginary components are shown in (c) and (d). The sum of the measurements enclosed by the red squares defines the expectation value of the identity (probability conservation).
Figure 15.
Figure 15. Concept diagram illustrating the measurements required on two photons, A and B, to perform an over-complete QST (highlighted in yellow) and a Bell violation measurement (highlighted in gray) for polarization and spatial mode correlations. The values show the expected outcomes for a biphoton maximally entangled state. By following the rows/columns through from one end to the other, one finds the equivalent measurement in the alternative basis, i.e., from polarization to spatial mode and vice versa. Measurements on hybrid states can be deduced by selecting a row of one degree of freedom and a column of another, i.e., photon A from the polarization and photon B from the OAM space. The spatial modes are shown with their phases as insets, the latter actually used in the measurement process.
Figure 16.
Figure 16. Each entangled photon in the pair is directed to a SLM that displays a hologram. The hologram is programmed with appropriate phase patterns to detect spatial modes. In this example, the holograms are shown for OAM modes of | | = ± 3 as well as the superpositions thereof, all six required for over-complete tomographic reconstruction. Running through six holograms on each SLM produces the 36 measurements required.
Figure 17.
Figure 17. Examples of QST measurements and the resulting density matrices for hybrid (left) and OAM (middle and right) biphoton states, with varying degrees of entanglement.
Figure 18.
Figure 18. Description of hybrid states on the HOPS. States on the surface of the HOPS are constructed from the tensor product of OAM states from the Bloch sphere and polarization states from the Poincaré sphere. The entire space is four dimensional but illustrated here by spheres representing the subspaces. The equator represents maximally entangled states, whereas the poles represent non-entangled states.
Figure 19.
Figure 19. (a) Density matrix representation for a maximally entangled biphoton hybrid state and similarly in (b) for a hybrid state that is not entangled.
Figure 20.
Figure 20. Top: conventional quantum experiment with biphotons using an SPDC source and projections using SLMs to explore spatial mode entanglement. Here two photons are produced at the crystal and travel in equal but opposite directions due to the crystal phase-matching condition. Middle: the detector in arm A is replaced with a source of bright light. The light travels backward through the system, bouncing off the crystal and passing through arm B to the detector. Because the angle of incidence equals the angle of reflection, the light in arm B has the correct properties to mimic the SPDC photon in this arm. Bottom: this concept can be further extended to simulate different SPDC processes by replacing the mirror by a third SLM, i.e., to simulate the mode of the pump beam.
Figure 21.
Figure 21. In the usual SPDC process the OAM modes in arms A and B have opposite sign due to conservation of OAM (assuming a Gaussian pump mode). In the classical experiment with one detector replaced by a laser, the extra reflection off the mirror suffices to flip the sign of the OAM that travels to arm B, thus mimicking the physics correctly for a QST.
Figure 22.
Figure 22. OAM spiral bandwidth shown for (a) the SPDC experiment with single photons (left) and the classical backprojection experiment (right). In (b) we show the outcome of a full QST (with differing degrees of freedom), again with quantum on the left and classical on the right.
Figure 23.
Figure 23. Example of actual classical signals at the detector (left) together with the resulting full QST on the classical beam (middle) and the corresponding quantum case (right). The agreement is excellent. Here the QST was performed on a d = 3 Hilbert space.
Figure 24.
Figure 24. The backprojection approach can also be used to demonstrate other quantum experiments, for example, ghost imaging. Here we illustrated examples of ghost imaging with position and momentum correlations using SPDC photons as well as a classically equivalent backprojection experiment. Copyright 2014 from “Experimental demonstration of Klyshko’s advanced-wave picture using a coincidence-count based, camera-enabled imaging system,” Aspden et al. [119]. Reproduced by permission of Taylor and Francis Group, LLC, a division of Informapic.
Figure 25.
Figure 25. (a) Illustrative experiment able to reveal the separability and non-separability of vector and scalar modes, consisting of a light source, a vector/scalar beam generator, an adjustable polarizer, and a spatially resolved camera detector. (b) Intensity and polarization map of a few vector beams. The non-separability of the space and polarization degrees of freedom is reflected in the space variant (inhomogeneous) polarization. (c) The non-separability of the two degrees of freedom manifests itself if one passes the beam through an adjustable polarizer and measures the resulting spatial mode on a camera as the polarizer is rotated, as represented here for two vector beams.
Figure 26.
Figure 26. (a) Representation of a SOC based on a dielectric metasurface for visible light. This device converts a left-circularly polarized light Gaussian beam into a right-circularly polarized helical beam with OAM m = 2 . (b) Design of a rectangular-section dielectric nanopost (nanofin). If the material is TiO 2 , and the height of the post ( H ) is 600 nm, in order to achieve structural birefringence value of π at 532 nm incident wavelength, the width (W) and the length (L) must be, respectively, 90 and 250 nm. (c) SEM micrograph of a TiO 2 metasurface SOC that produces helical beams of OAM ± 1 . (d) Beam profile generated by the device in (c). (e) and (f) Interference of the helical beam in (d) with a reference beam. The spiral interference handedness depends on the sign of the helical beam OAM.
Figure 27.
Figure 27. (a) SEM image of a J -plate. This dielectric metasurface device is made of different and differently oriented nanoposts with the same height. This allows to control both the propagation phase and the geometrical phase to map the spin base into an arbitrary subspace of the OAM. For instance, as represented on the HOPS, left-circularly polarized light is converted into right-circularly polarized light with OAM + 3 , while right-circularly polarized light is converted into left-circularly polarized beam with OAM of + 6 , shown in (b) and (c). When the input polarization state is not a pure state, both helical beams are generated. These two beams are in orthogonal polarization states and do not interfere. However, when they are both projected into the same polarization state by means of a polarizer, they produce complex interference pictures. (d) The interference picture obtained from incident horizontal polarization state. (e) The same as (d) when the incident light is vertically polarized.
Figure 28.
Figure 28. Graphical representation of the tomographic projections onto both eigenstates of the OAM and polarization Pauli matrices for (a) a maximally non-separable vector mode and (b) a left-circularly polarized OAM mode with = 1 (scalar mode). (c) and (d) show the reconstructed density matrices for the classical vector and scalar modes in (a) and (b), respectively.
Figure 29.
Figure 29. Bell measurement with spatially separated photon pairs generated from a spontaneous parametric downconversion source. Local measurements were performed using spatial analyzers that project each photon onto superposition states defined on the equator of the Bloch sphere. The projections were mapped onto the states ( | + exp ( 2 i θ 1 ) | ) / 2 and ( | + exp ( 2 i θ 2 ) | ) / 2 . Subsequently, the signals of the projected photons were measured in coincidence with an amplitude proportional to I ( θ 1 , θ 2 ) . The high visibility of the amplitude variation is indicative of non-local interactions between the spatially separated photons [208].
Figure 30.
Figure 30. Simulation of Bell measurements on an input vector vortex beam. The angles θ 1 and θ 2 parameterize the states used to compute the projection. These are polarization and OAM superposition states on the equator of the Poincaré and Bloch spheres, ( | R + exp ( 2 i θ 1 ) | L ) / 2 and ( | + exp ( 2 i θ 2 ) | ) / 2 .
Figure 31.
Figure 31. Experimental setup scheme: (a) state generation and (b) automated state projection. LS, laser source (532 nm fiber-coupled light from Verdi G5, Coherent); L i , Fourier lenses; P 0 , 1 & 2 , polarizers; HWP 0 , 1 & 2 , half-waveplates; QWP 0 , 1 & 2 , quarter-waveplates; CCD, Chameleon3 CCD camera (Point Grey, FLIR). The waveplates HWP 1 & 2 and QWP 1 & 2 are mounted on roto-flip mounts. The lenses L 1 ( f = 300 mm ) and L 2 ( f = 50 mm ) demagnify the laser beam to match the size of J 1 , L 3 & 4 ( f = 300 mm ) relay the plane of J 1 onto J 2 , L 5 ( f = 150 mm ) Fourier transforms the J 2 plane to spatially project into the Gaussian mode ( = 0 ) or, replacing it by L 6 & 7 ( f = 75 mm ), the plane of J 2 is relayed onto the CCD for alignment purposes. Polarizers P 1 & 2 are polarizing beam splitters, providing extra output from port orthogonal to the optical axis for convenience during alignment.
Figure 32.
Figure 32. Thirty-six angular arrangements of the polarization optics. The subscript-1 and subscript-2 terms indicate the polarization states and the OAM states created with the first and second pairs of QWPs and HWPs, respectively.
Figure 33.
Figure 33. In our experimental realization, the indicated angular positions in radians of QWP 1 , QWP 1 and QWP 2 , HWP 2 allow to generate the desired polarization states. The subscript-1 and subscript-2 terms indicate the polarization states and the OAM states created with the first and second pairs of QWPs and HWPs, respectively.
Figure 34.
Figure 34. Quantum-state tomography measurements (a) theory and (b) experiment, with resulting density matrices in (c) and (d), respectively, for a scalar mode of the form | ψ 0 = | = 1 | R .
Figure 35.
Figure 35. Quantum-state tomography measurements (a) theory and (b) experiment, with resulting density matrices in (c) and (d), respectively, for a horizontally polarized mode of the form | ψ 0 = ( | = 1 + | = 1 ) | H .
Figure 36.
Figure 36. Quantum-state tomography (a) theory and (b) experiment, with resulting density matrices in (c) and (d), respectively, for a vector mode of the form | ψ 0 = 1 2 ( | = 1 | R + | = 1 | L ) .
Figure 37.
Figure 37. Quantum-state tomography (a) theory and (b) experiment, with resulting density matrices in (c) and (d), respectively, for a vector mode | ψ 0 = 1 2 ( | = 1 | R + i | = 1 | L ) .
Figure 38.
Figure 38. Experimental Bell measurement curves for (a), (b) two “classically entangled” vector modes, and (c), (d) two “non-entangled” scalar modes. The solid lines represent the expected theoretical values, whereas the scattered data points represent the experimentally measured values.
Figure 39.
Figure 39. Detailed description of the 3D-printed roto-flip stage. (a) Front and back views; (b) 3D-printed parts assembly. The numbers in the figure indicate the following: (1) roto-flip stage support board; (2) parallax standard servo; (3) servo connection board; (4) 28BYJ-48 stepper motor; (5) back-support board for stepper with ball-bearing slot; (6) small spur gear with 26 teeth; (7) ball bearing (17 mm inside diameter, 26 mm outside diameter, 5 mm race width); (8) big spur gear with 104 teeth and ball-bearing slot; (9) clip for SM1 optical components. The various 3D printed parts are assembled by both pressure fitting and by using common M4 screws and nuts. (4), (5), (6), (7), and (8) can be pressure fitted. The screw holes in (1) and (5) were threaded with a tapping tool to complete the assembly.
Figure 40.
Figure 40. Electronics schematic diagram. The numbers in the figure indicate the following: (1) 28BYJ-48 stepper motor; (2) 1 A, 12 V power supply socket; (3) ULN200xx chip stepper-driver board; (4) Arduino Nano microcontroller; (5) parallax standard servo.
Figure 41.
Figure 41. Video demonstration of the automated state-tomography system. (a)–(c) Three angular arrangements of the polarization optics as performed during a tomographic measurement are shown. In the video (see Visualization 1 [126]), we show our automated tomography system in action: both the projected states, as acquired by the CCD camera, and the extracted intensities are displayed, as the system iterates through the 36 programmed angular positions of the polarization optics.

Tables (7)

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Table 1. Linear Polarization States Produced for θ = π / 2

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Table 2. Eigenvectors and Eigenvalues of the Identity and Pauli Matrices in the Polarization Basis

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Table 3. Tomographic Measurements of a Pure State

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Table 4. Tomographic Measurements of a Mixed State

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Table 5. Transformation of an Input Horizontally Polarized State on the Surface of the Poincaré Sphere Using Wave Plates

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Table 6. Spin-Orbit States Produced through Polarization Control

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Table 7. Post-Selected Horizontally Polarized State

Equations (79)

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| ψ = α | 0 + β | 1 ,
| ψ = cos ( θ / 2 ) | R + exp ( i ϕ ) sin ( θ / 2 ) | L ,
| ψ | ψ | 2 = | cos 2 ( θ / 2 ) | 2 + sin 2 ( θ / 2 ) = 1 .
ρ mixed = m c m | ψ m ψ m | .
ρ = | ψ ψ | = cos 2 ( θ / 2 ) | R R | + sin 2 ( θ / 2 ) | L L | + cos ( θ / 2 ) sin ( θ / 2 ) exp ( i ϕ ) | R L | + cos ( θ / 2 ) sin ( θ / 2 ) exp ( i ϕ ) | L R | .
ρ = ( cos 2 ( θ / 2 ) cos ( θ / 2 ) sin ( θ / 2 ) exp ( i ϕ ) cos ( θ / 2 ) sin ( θ / 2 ) exp ( i ϕ ) sin 2 ( θ / 2 ) ) .
tr ( ρ ) = cos 2 ( θ / 2 ) + sin 2 ( θ / 2 ) = 1 .
tr ( ρ 2 ) = tr ( m n c m c n | ψ m ψ m | | ψ n ψ n | ) = m n c m c n | ψ m | ψ n | 2 .
0 m n c m c n | ψ m | ψ n | 2 m c m n c n = 1 ,
{ tr ( ρ 2 ) = 1 for c m = c n = 1 ρ is pure 0 tr ( ρ 2 ) < 1 for c m , c n < 1 ρ is mixed .
σ 1 = ( 0 1 1 0 ) ; σ 2 = ( 0 i i 0 ) ; σ 3 = ( 1 0 0 1 ) ; I = ( 1 0 0 1 ) .
tr ( σ i σ j ) = 2 δ i , j , for i , j = 1 , 2 or 3 .
I = σ 0 = ( 1 0 0 1 ) ; tr ( σ 0 σ 0 ) = 2 ; tr ( σ 0 σ i ) = tr ( σ i σ 0 ) = tr ( σ j ) = 0 .
ρ = 1 2 n = 0 3 ρ n σ n ,
tr ( ρ σ n ) = 1 2 tr ( m = 0 3 ρ m σ m σ n ) = 1 2 m = 0 3 ρ m tr ( σ m σ n ) = m = 0 3 ρ m δ m , n = ρ n .
σ n = α n 0 | λ n 0 λ n 0 | + α n 1 | λ n 1 λ n 1 | .
ρ n = tr ( ρ σ n ) = α n 0 λ n 0 | ρ | λ n 0 + α n 1 λ n 1 | ρ | λ n 1 .
ρ 0 = tr ( ρ σ 0 ) = R | ρ | R + L | ρ | L ,
ρ 1 = tr ( ρ σ 1 ) = H | ρ | H V | ρ | V ,
ρ 2 = tr ( ρ σ 2 ) = D | ρ | D A | ρ | A ,
ρ 3 = tr ( ρ σ 3 ) = R | ρ | R L | ρ | L .
| ψ = 3 2 | R + 1 2 exp ( i π 3 ) | L ,
ρ = ( 3 4 3 4 exp ( i π 3 ) 3 4 exp ( i π 3 ) 1 4 ) .
ρ 0 = 1 ; ρ 1 = 3 4 ; ρ 2 = 3 4 ; ρ 3 = 1 2 ,
ρ = 1 2 ( 1 0 0 1 ) + 3 8 ( 0 1 1 0 ) 3 8 ( 0 i i 0 ) + 1 4 ( 1 0 0 1 ) = ( 3 4 3 + 3 i 8 3 3 i 8 1 4 ) = ( 3 4 3 4 exp ( i π 3 ) 3 4 exp ( i π 3 ) 1 4 ) .
ρ = 1 3 | R R | + 2 3 | L L | = ( 1 3 0 0 2 3 ) .
ρ 0 = 1 ; ρ 1 = 0 ; ρ 2 = 0 ; ρ 3 = 1 3 ,
ρ = 1 2 ( 1 0 0 1 ) 1 6 ( 1 0 0 1 ) = ( 1 3 0 0 2 3 ) .
| ψ = cos ( θ / 2 ) | 1 + exp ( i ϕ ) sin ( θ / 2 ) | 2 ,
| A ( r , z ) exp ( i ϕ ) ,
1 | 2 = 0 d r r A ( r , z 0 ) 0 2 π <