Abstract

The ability to completely characterize the state of a system is an essential element for the emerging quantum technologies. Here, we present a compressed-sensing-inspired method to ascertain any rank-deficient qudit state, which we experimentally encode in photonic orbital angular momentum. We efficiently reconstruct these qudit states from a few scans with an intensified CCD camera. Since it only requires a small number of intensity measurements, our technique provides an easy and accurate way to identify quantum sources, channels, and systems.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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).
    [Crossref] [PubMed]
  2. 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).
    [Crossref]
  3. A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, 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).
    [Crossref]
  4. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
    [Crossref] [PubMed]
  5. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
    [Crossref]
  6. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
    [Crossref] [PubMed]
  7. 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).
    [Crossref] [PubMed]
  8. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
    [Crossref] [PubMed]
  9. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009).
    [Crossref]
  10. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
    [Crossref]
  11. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
    [Crossref] [PubMed]
  12. N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).
  13. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
    [Crossref]
  14. E. J. Candes and T. Tao, “The power of convex relaxation: Near-optimal matrix completion,” IEEE Trans. Inf. Theory 56, 2053–2080 (2010).
    [Crossref]
  15. Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications(Cambridge University, 2012).
    [Crossref]
  16. A. Stern, Optical Compressive Imaging (CRC, 2016).
  17. 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).
    [Crossref]
  18. A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
    [Crossref] [PubMed]
  19. W.-T. Liu, T. Zhang, J.-Y. Liu, P.-X. Chen, and J.-M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108, 170403 (2012).
    [Crossref] [PubMed]
  20. C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
    [Crossref] [PubMed]
  21. A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
    [Crossref]
  22. F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
    [Crossref] [PubMed]
  23. A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
    [Crossref]
  24. C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
    [Crossref] [PubMed]
  25. A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj quantum inf. 1, 15018 (2015).
    [Crossref]
  26. A. Peres, Quantum Theory: Concepts and Methods(Kluwer, 2002).
  27. E. Prugovečki, “Information-theoretical aspects of quantum measurement,” Int. J. Theor. Phys. 16, 321–331 (1977).
    [Crossref]
  28. P. Busch and P. J. Lahti, “The determination of the past and the future of a physical system in quantum mechanics,” Found. Phys. 19, 633–678 (1989).
    [Crossref]
  29. E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
    [Crossref]
  30. S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).
    [Crossref]
  31. A. Siegman, Lasers(Oxford University, 1986).
  32. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  33. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications(Wiley, 1977).
  34. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063–1068 (2007).
    [Crossref]
  35. 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).
    [Crossref] [PubMed]

2017 (3)

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, 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).
[Crossref]

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

2015 (3)

A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj quantum inf. 1, 15018 (2015).
[Crossref]

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).
[Crossref]

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

2014 (4)

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).
[Crossref] [PubMed]

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref] [PubMed]

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

2013 (2)

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).
[Crossref] [PubMed]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

2012 (2)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

W.-T. Liu, T. Zhang, J.-Y. Liu, P.-X. Chen, and J.-M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108, 170403 (2012).
[Crossref] [PubMed]

2011 (1)

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

2010 (3)

E. J. Candes and T. Tao, “The power of convex relaxation: Near-optimal matrix completion,” IEEE Trans. Inf. Theory 56, 2053–2080 (2010).
[Crossref]

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).
[Crossref]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

2009 (1)

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009).
[Crossref]

2008 (1)

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
[Crossref]

2007 (1)

2006 (1)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[Crossref]

2004 (2)

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

2002 (1)

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).
[Crossref] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

1989 (1)

P. Busch and P. J. Lahti, “The determination of the past and the future of a physical system in quantum mechanics,” Found. Phys. 19, 633–678 (1989).
[Crossref]

1977 (1)

E. Prugovečki, “Information-theoretical aspects of quantum measurement,” Int. J. Theor. Phys. 16, 321–331 (1977).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Agnew, M.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Almeida, M. P.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Barends, R.

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Becker, S.

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).
[Crossref]

Beijersbergen, M.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

Bell, B. A.

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

Ben-Israel, A.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications(Wiley, 1977).

Bent, N.

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

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).
[Crossref] [PubMed]

Berkhout, G. C. G.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

Blatt, R.

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

Bolduc, E.

Bouchard, F.

Boyd, R. W.

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, 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).
[Crossref]

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).
[Crossref]

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).
[Crossref]

Broome, M. A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Busch, P.

P. Busch and P. J. Lahti, “The determination of the past and the future of a physical system in quantum mechanics,” Found. Phys. 19, 633–678 (1989).
[Crossref]

Candes, E. J.

E. J. Candes and T. Tao, “The power of convex relaxation: Near-optimal matrix completion,” IEEE Trans. Inf. Theory 56, 2053–2080 (2010).
[Crossref]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[Crossref]

Candès, E. J.

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
[Crossref]

Chan, S.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

Chen, P.-X.

W.-T. Liu, T. Zhang, J.-Y. Liu, P.-X. Chen, and J.-M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108, 170403 (2012).
[Crossref] [PubMed]

Courtial, J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

D’Ambrosio, V.

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).
[Crossref] [PubMed]

Deutsch, I. H.

A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj quantum inf. 1, 15018 (2015).
[Crossref]

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Eisert, J.

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

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).
[Crossref]

Eldar, Y. C.

Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications(Cambridge University, 2012).
[Crossref]

Elser, D.

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Fedrizzi, A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Fickler, R.

Flammia, S. T.

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

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).
[Crossref]

Franke-Arnold, S.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Gagnon-Bischoff, J.

Gauthier, D. J.

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).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gibson, G.

Greville, T. N. E.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications(Wiley, 1977).

Gross, D.

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref] [PubMed]

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).
[Crossref]

Gühne, O.

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref] [PubMed]

Günthner, K.

Heim, B.

Heshami, K.

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Kalev, A.

A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj quantum inf. 1, 15018 (2015).
[Crossref]

Karimi, E.

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, 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).
[Crossref]

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

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).
[Crossref] [PubMed]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009).
[Crossref]

Kelly, J.

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

Korotkov, A. N.

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

Kosut, R. L.

A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj quantum inf. 1, 15018 (2015).
[Crossref]

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Kutyniok, G.

Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications(Cambridge University, 2012).
[Crossref]

Lahti, P. J.

P. Busch and P. J. Lahti, “The determination of the past and the future of a physical system in quantum mechanics,” Found. Phys. 19, 633–678 (1989).
[Crossref]

Larocque, H.

Lavery, M. P. J.

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).
[Crossref]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

Leach, J.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Leuchs, G.

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, 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).
[Crossref]

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

Lindsay, A.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

Liu, J.-Y.

W.-T. Liu, T. Zhang, J.-Y. Liu, P.-X. Chen, and J.-M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108, 170403 (2012).
[Crossref] [PubMed]

Liu, W.-T.

W.-T. Liu, T. Zhang, J.-Y. Liu, P.-X. Chen, and J.-M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108, 170403 (2012).
[Crossref] [PubMed]

Liu, Y.-K.

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).
[Crossref]

Luis, A.

Magaña-Loaiza, O. S.

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).
[Crossref]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Malik, M.

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).
[Crossref]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

Marquardt, C.

Marrucci, L.

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).
[Crossref] [PubMed]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009).
[Crossref]

Martinis, J. M.

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

McCutcheon, W.

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

McMillan, A.

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

Mirhosseini, M.

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).
[Crossref]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

Mohseni, M.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Monz, T.

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

Moroder, T.

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref] [PubMed]

Nagali, E.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009).
[Crossref]

Nigg, D.

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

Niggebaum, A.

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref] [PubMed]

O’Sullivan, M. N.

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).
[Crossref]

Padgett, M. J.

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).
[Crossref]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Pas’ko, V.

Peres, A.

A. Peres, Quantum Theory: Concepts and Methods(Kluwer, 2002).

Peuntinger, C.

Piccirillo, B.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009).
[Crossref]

Prugovecki, E.

E. Prugovečki, “Information-theoretical aspects of quantum measurement,” Int. J. Theor. Phys. 16, 321–331 (1977).
[Crossref]

Qassim, H.

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

Rabitz, H.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Rarity, J. G.

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Riofrío, C. A.

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

Rodenburg, B.

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).
[Crossref]

Rodionov, A. V.

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[Crossref]

Roth, I.

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

Sánchez-Soto, L. L.

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

Sank, D.

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

Santamato, E.

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).
[Crossref] [PubMed]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schwemmer, C.

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref] [PubMed]

Sciarrino, F.

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).
[Crossref] [PubMed]

Shabani, A.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Shi, Z.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

Siegman, A.

A. Siegman, Lasers(Oxford University, 1986).

Sit, A.

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

Slussarenko, S.

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).
[Crossref] [PubMed]

Sponselli, A.

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).
[Crossref] [PubMed]

Steffens, A.

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

Stern, A.

A. Stern, Optical Compressive Imaging (CRC, 2016).

Sych, D.

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

Tahir, A. A.

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

Tame, M. S.

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

Tao, T.

E. J. Candes and T. Tao, “The power of convex relaxation: Near-optimal matrix completion,” IEEE Trans. Inf. Theory 56, 2053–2080 (2010).
[Crossref]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[Crossref]

Tonolini, F.

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

Tóth, G.

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref] [PubMed]

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Vallone, G.

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).
[Crossref] [PubMed]

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).
[Crossref]

Vasnetsov, M.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Veitia, A.

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

Villoresi, P.

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).
[Crossref] [PubMed]

Wang, J.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Weinfurter, H.

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref] [PubMed]

Wenner, J.

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

White, A. G.

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

Willner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Yuan, J.-M.

W.-T. Liu, T. Zhang, J.-Y. Liu, P.-X. Chen, and J.-M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108, 170403 (2012).
[Crossref] [PubMed]

Yue, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

Zhang, T.

W.-T. Liu, T. Zhang, J.-Y. Liu, P.-X. Chen, and J.-M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108, 170403 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94, 231124 (2009).
[Crossref]

C. R. Math. (1)

E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” C. R. Math. 346, 589–592 (2008).
[Crossref]

Found. Phys. (1)

P. Busch and P. J. Lahti, “The determination of the past and the future of a physical system in quantum mechanics,” Found. Phys. 19, 633–678 (1989).
[Crossref]

IEEE Trans. Inf. Theory (1)

E. J. Candes and T. Tao, “The power of convex relaxation: Near-optimal matrix completion,” IEEE Trans. Inf. Theory 56, 2053–2080 (2010).
[Crossref]

IEEE Trans. Inform. Theory (1)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[Crossref]

Int. J. Theor. Phys. (1)

E. Prugovečki, “Information-theoretical aspects of quantum measurement,” Int. J. Theor. Phys. 16, 321–331 (1977).
[Crossref]

J. Opt. Soc. Am. A (1)

Nat. Commun. (2)

C. A. Riofrío, D. Gross, S. T. Flammia, T. Monz, D. Nigg, R. Blatt, and J. Eisert, “Experimental quantum compressed sensing for a seven-qubit system,” Nat. Commun. 8, 15305 (2017).
[Crossref] [PubMed]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref] [PubMed]

Nat. Photonics (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012).
[Crossref]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313 (2001).
[Crossref] [PubMed]

New J. Phys. (1)

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).
[Crossref]

npj quantum inf. (1)

A. Kalev, R. L. Kosut, and I. H. Deutsch, “Quantum tomography protocols with positivity are compressed sensing protocols,” npj quantum inf. 1, 15018 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optica (1)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. B (1)

A. V. Rodionov, A. Veitia, R. Barends, J. Kelly, D. Sank, J. Wenner, J. M. Martinis, R. L. Kosut, and A. N. Korotkov, “Compressed sensing quantum process tomography for superconducting quantum gates,” Phys. Rev. B 90, 144504 (2014).
[Crossref]

Phys. Rev. Lett. (8)

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).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref] [PubMed]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

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).
[Crossref]

A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A. Broome, M. P. Almeida, A. Fedrizzi, and A. G. White, “Efficient measurement of quantum dynamics via compressive sensing,” Phys. Rev. Lett. 106, 100401 (2011).
[Crossref] [PubMed]

W.-T. Liu, T. Zhang, J.-Y. Liu, P.-X. Chen, and J.-M. Yuan, “Experimental quantum state tomography via compressed sampling,” Phys. Rev. Lett. 108, 170403 (2012).
[Crossref] [PubMed]

C. Schwemmer, G. Tóth, A. Niggebaum, T. Moroder, D. Gross, O. Gühne, and H. Weinfurter, “Experimental comparison of efficient tomography schemes for a six-qubit state,” Phys. Rev. Lett. 113, 040503 (2014).
[Crossref] [PubMed]

Phys. Rev. X (1)

N. Bent, H. Qassim, A. A. Tahir, D. Sych, G. Leuchs, L. L. Sánchez-Soto, E. Karimi, and R. W. Boyd, “Experimental realizationof quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures,” Phys. Rev. X 5, 041006 (2015).

Quantum Sci. Technol. (1)

A. Steffens, C. A. Riofrío, W. McCutcheon, I. Roth, B. A. Bell, A. McMillan, M. S. Tame, J. G. Rarity, and J. Eisert, “Experimentally exploring compressed sensing quantum tomography,” Quantum Sci. Technol. 2, 025005 (2017).
[Crossref]

Sci. Rep. (1)

F. Tonolini, S. Chan, M. Agnew, A. Lindsay, and J. Leach, “Reconstructing high-dimensional two-photon entangled states via compressive sensing,” Sci. Rep. 4, 6542 (2014).
[Crossref] [PubMed]

Other (6)

Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications(Cambridge University, 2012).
[Crossref]

A. Stern, Optical Compressive Imaging (CRC, 2016).

A. Peres, Quantum Theory: Concepts and Methods(Kluwer, 2002).

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications(Wiley, 1977).

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).
[Crossref]

A. Siegman, Lasers(Oxford University, 1986).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 Simulation of our compressed sensing protocol with twisted photons. (a) Number of independent detections nZ generated by Z intensity scans along the propagation axis performed on a signal with max = 7(d = 15). The maximum of nZ = 218 detections is obtained for Z ≥ 8 scans. (b) Reconstructions errors, ε ¯, from the compressed sensing protocol of the twisted photons as a function of the rank of the state for a fixed dimension (max = 7) and for Z = 1, 2, and 3 CCD scans, with and without positivity constraint. (c) The reconstructions errors for two CCD scans and different dimensions. In all the cases, we take a 19 × 19 pixels screen.
Fig. 2
Fig. 2 Sketch of the experimental setup. A photonic state is generated by manipulating the phase and intensity of an incoming beam via the spatial light modulator (SLM). The beam is then focused using a variable holographic lens imprinted on the SLM together with the state generation hologram. The ICCD camera has a fixed position and records intensity scans. Inset shows the state and lens phase patterns, [0, 2π), in a hue color.
Fig. 3
Fig. 3 Experimental reconstruction from compressed sensing. (a) Density matrix of the true state and (b) Reconstructed density matrix after two intensity scans with a signal space spanned by ∈ {−4,…, 4}. The upper row shows four experimental ICCD scans at the planes z/zR =0, 1/3, 1/2, and 1, respectively. The lower row shows the predictions from the reconstructed state of the same ICCD scans.
Fig. 4
Fig. 4 Experimental reconstruction from compressed sensing. (a) Density matrix of the true state and (b) Reconstructed density matrix after one intensity scan. The upper row shows four experimental ICCD scans at the planes z/zR =0, 1/3, 1/2, and 1, respectively. The lower row shows the predictions from the reconstructed state of the same ICCD scans.
Fig. 5
Fig. 5 Entropy S as a function of the number of intensity scans Z, for a signal space spanned by ∈ {−4,…, 4} and for true states of the form in (10). The circles are obtained by averaging over 20 random states and the error bars indicate the corresponding variance. Blue refers to protocol with positivity, while red is without positivity.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

ϱ ^ = min ϱ A [ ϱ ] p subject to ϱ 0 ,
LG p ( r , ϕ , z ) = r , ϕ , z | , p = 2 p ! π ( p + | | ) ! 1 w ( z ) ( 2 r w ( z ) ) | | × L p | | ( 2 r 2 w ( z ) 2 ) exp ( r 2 [ 1 w ( z ) 2 i k 2 R ( z ) ] i ϕ i ψ p ( z ) ) ,
w 2 ( z ) = w 0 2 [ 1 + ( z / z R ) 2 ] , R ( z ) = z [ 1 + ( z R / z ) 2 ] , ψ p ( z ) = ( 2 p + | | + 1 ) arctan ( z / z R ) ,
p ( r , ϕ , z ) = r , ϕ , z | ϱ | r , ϕ , z = , LG , 0 ( r , ϕ , z ) ϱ LG , 0 * ( r , ϕ , z ) e 2 r 2 ϱ C ,
C = r | | + | | exp [ i ( ) ϕ ] exp { i [ ψ ( z ) ψ ( z ) ] } , r ( z ) = r / w ( z ) .
| | | | = N , = M , | | | | = s ,
n Z 4 max   2 + 4 max + 1 = d 2 1 2 ( d 1 ) .
n Z = 1 = max   2 + 4 max + 1 = 1 4 ( d 2 + 6 d 3 )
n Z = 2 = 2 max   2 + 7 max 1 = 1 2 ( d 2 + 5 d 8 ) .
ϱ = p | 0 0 | + ( 1 p ) | Ψ Ψ | ,