Abstract

We present the experimental quantum tomography of 7- and 8-dimensional quantum systems based on projective measurements in the mutually unbiased basis (MUB-QT). One of the advantages of MUB-QT is that it requires projections from a minimal number of bases to be performed. In our scheme, the higher dimensional quantum systems are encoded using the propagation modes of single photons, and we take advantage of the capabilities of amplitude- and phase-modulation of programmable spatial light modulators to implement the MUB-QT.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957).
    [CrossRef]
  2. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
    [CrossRef]
  3. R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
    [CrossRef]
  4. 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).
    [CrossRef]
  5. W. K. Wootters, and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989).
    [CrossRef]
  6. R. B. A. Adamson, and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
    [CrossRef] [PubMed]
  7. H. Haffner, W. Hansel, C. F. Roos, and J. Benhelm, “D. Chek-al-kar, M. Chwalla, T. Korber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne,W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
    [CrossRef] [PubMed]
  8. N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
    [CrossRef] [PubMed]
  9. J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Photonic State Tomography, Advances in AMO Physics (Elsevier, 2006), Vol. 52, Chap. 3.
  10. I. D. Ivanovic, “Geometrical description of quantal state determination,” J. Phys. A: Math. Theor. 14, 3241–3245 (1981).
    [CrossRef]
  11. A. B. Klimov, C. Muoz, A. Fern’andez, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303 (2008).
    [CrossRef]
  12. T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
    [CrossRef]
  13. D. Kaszlikowski, P. Gnaci’nski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubit,” Phys. Rev. Lett. 85, 4418–4421 (2000).
    [CrossRef] [PubMed]
  14. L. Neves, G. Lima, J. G. A. G’omez, C. H. Monken, C. Saavedra, and S. P’adua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
    [CrossRef] [PubMed]
  15. M. N. O. Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).
  16. G. Lima, A. Vargas, L. Neves, R. Guzm’an, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express 17, 10688–10696 (2009).
    [CrossRef] [PubMed]
  17. R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
    [CrossRef]
  18. G. Lima, L. Neves, I. F. Santos, J. G. Aguirre G’omez, C. Saavedra, and S. P’adua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
    [CrossRef]
  19. G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. P’adua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
    [CrossRef]
  20. G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
    [CrossRef]
  21. M. S. Kaznady, and D. F. V. James, “Numerical strategies for quantum tomography: alternatives to full optimization,” Phys. Rev. A 79, 022109 (2009).
    [CrossRef]
  22. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
    [CrossRef]
  23. T. O. Maciel, and R. O. Vianna, “Viable entanglement detection of unknown mixed states in low dimensions,” Phys. Rev. A 80, 032325 (2009).
    [CrossRef]
  24. G. Lima, E. S. G’omez, A. Vargas, R. O. Vianna, and C. Saavedra, “Fast entanglement detection for unknown states of two spatial qutrits,” Phys. Rev. A 82, 012302 (2010).
    [CrossRef]
  25. R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
    [CrossRef]
  26. S. P. Walborn, D. S. Lamelle, M. P. Almeida, and P. H. Souto Ribeiro, “Quantum key distribution with higherorder alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
    [CrossRef] [PubMed]
  27. G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
    [CrossRef]
  28. J. L. Romero, G. Bork, A. B. Klimov, and L. L. S’anchez-Soto, “Structure of the sets of mutually unbiased bases for N qubits,” Phys. Rev. A 72, 062310 (2005).
    [CrossRef]
  29. J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2181 (2004).
    [CrossRef]
  30. I. Sainz, L. Roa, and A. B. Klimov, “Unbiased nonorthogonal bases for tomographic reconstruction,” Phys. Rev. A 81, 052114 (2010).
    [CrossRef]
  31. C. Paiva, E. Burgos-Inostroza, O. Jim’enez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A 82, 032115 (2010).
    [CrossRef]

2010 (4)

R. B. A. Adamson, and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
[CrossRef] [PubMed]

G. Lima, E. S. G’omez, A. Vargas, R. O. Vianna, and C. Saavedra, “Fast entanglement detection for unknown states of two spatial qutrits,” Phys. Rev. A 82, 012302 (2010).
[CrossRef]

I. Sainz, L. Roa, and A. B. Klimov, “Unbiased nonorthogonal bases for tomographic reconstruction,” Phys. Rev. A 81, 052114 (2010).
[CrossRef]

C. Paiva, E. Burgos-Inostroza, O. Jim’enez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A 82, 032115 (2010).
[CrossRef]

2009 (3)

G. Lima, A. Vargas, L. Neves, R. Guzm’an, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express 17, 10688–10696 (2009).
[CrossRef] [PubMed]

T. O. Maciel, and R. O. Vianna, “Viable entanglement detection of unknown mixed states in low dimensions,” Phys. Rev. A 80, 032325 (2009).
[CrossRef]

M. S. Kaznady, and D. F. V. James, “Numerical strategies for quantum tomography: alternatives to full optimization,” Phys. Rev. A 79, 022109 (2009).
[CrossRef]

2008 (4)

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).
[CrossRef]

A. B. Klimov, C. Muoz, A. Fern’andez, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303 (2008).
[CrossRef]

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. P’adua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
[CrossRef]

2006 (2)

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre G’omez, C. Saavedra, and S. P’adua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
[CrossRef]

S. P. Walborn, D. S. Lamelle, M. P. Almeida, and P. H. Souto Ribeiro, “Quantum key distribution with higherorder alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[CrossRef] [PubMed]

2005 (4)

J. L. Romero, G. Bork, A. B. Klimov, and L. L. S’anchez-Soto, “Structure of the sets of mutually unbiased bases for N qubits,” Phys. Rev. A 72, 062310 (2005).
[CrossRef]

L. Neves, G. Lima, J. G. A. G’omez, C. H. Monken, C. Saavedra, and S. P’adua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef] [PubMed]

M. N. O. Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).

H. Haffner, W. Hansel, C. F. Roos, and J. Benhelm, “D. Chek-al-kar, M. Chwalla, T. Korber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne,W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[CrossRef] [PubMed]

2004 (4)

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[CrossRef]

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2181 (2004).
[CrossRef]

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

2002 (1)

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[CrossRef]

2001 (2)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[CrossRef]

2000 (1)

D. Kaszlikowski, P. Gnaci’nski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubit,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[CrossRef] [PubMed]

1994 (1)

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
[CrossRef]

1989 (1)

W. K. Wootters, and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989).
[CrossRef]

1981 (1)

I. D. Ivanovic, “Geometrical description of quantal state determination,” J. Phys. A: Math. Theor. 14, 3241–3245 (1981).
[CrossRef]

1957 (1)

U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957).
[CrossRef]

1935 (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Adamson, R. B. A.

R. B. A. Adamson, and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
[CrossRef] [PubMed]

Aguirre G’omez, J. G.

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre G’omez, C. Saavedra, and S. P’adua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
[CrossRef]

Almeida, M. P.

S. P. Walborn, D. S. Lamelle, M. P. Almeida, and P. H. Souto Ribeiro, “Quantum key distribution with higherorder alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[CrossRef] [PubMed]

Bartlett, S. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Benhelm, J.

H. Haffner, W. Hansel, C. F. Roos, and J. Benhelm, “D. Chek-al-kar, M. Chwalla, T. Korber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne,W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[CrossRef] [PubMed]

Blume-Kohout, R.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2181 (2004).
[CrossRef]

Bork, G.

J. L. Romero, G. Bork, A. B. Klimov, and L. L. S’anchez-Soto, “Structure of the sets of mutually unbiased bases for N qubits,” Phys. Rev. A 72, 062310 (2005).
[CrossRef]

Boyd, R. W.

M. N. O. Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).

Burgos-Inostroza, E.

C. Paiva, E. Burgos-Inostroza, O. Jim’enez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A 82, 032115 (2010).
[CrossRef]

Caves, C. M.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2181 (2004).
[CrossRef]

Chen, J. L.

T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[CrossRef]

Dalton, R. B.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

de Burgh, M. D.

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).
[CrossRef]

Delgado, A.

C. Paiva, E. Burgos-Inostroza, O. Jim’enez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A 82, 032115 (2010).
[CrossRef]

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. P’adua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

Doherty, A. C.

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).
[CrossRef]

Dougakiuchi, T.

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
[CrossRef]

Durt, T.

T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[CrossRef]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Fano, U.

U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957).
[CrossRef]

Fern’andez, A.

A. B. Klimov, C. Muoz, A. Fern’andez, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303 (2008).
[CrossRef]

Fields, B. D.

W. K. Wootters, and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989).
[CrossRef]

G’omez, E. S.

G. Lima, E. S. G’omez, A. Vargas, R. O. Vianna, and C. Saavedra, “Fast entanglement detection for unknown states of two spatial qutrits,” Phys. Rev. A 82, 012302 (2010).
[CrossRef]

G’omez, J. G. A.

L. Neves, G. Lima, J. G. A. G’omez, C. H. Monken, C. Saavedra, and S. P’adua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef] [PubMed]

Gilchrist, A.

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).
[CrossRef]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Gnaci’nski, P.

D. Kaszlikowski, P. Gnaci’nski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubit,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[CrossRef] [PubMed]

Guzm’an, R.

Haffner, H.

H. Haffner, W. Hansel, C. F. Roos, and J. Benhelm, “D. Chek-al-kar, M. Chwalla, T. Korber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne,W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[CrossRef] [PubMed]

Hale, M. N. O.

M. N. O. Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).

Hansel, W.

H. Haffner, W. Hansel, C. F. Roos, and J. Benhelm, “D. Chek-al-kar, M. Chwalla, T. Korber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne,W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[CrossRef] [PubMed]

Harvey, M. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Hofmann, H. F.

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
[CrossRef]

Howell, J. C.

M. N. O. Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).

Iemmi, C.

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Iinuma, M.

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
[CrossRef]

Ivanovic, I. D.

I. D. Ivanovic, “Geometrical description of quantal state determination,” J. Phys. A: Math. Theor. 14, 3241–3245 (1981).
[CrossRef]

James, D. F. V.

M. S. Kaznady, and D. F. V. James, “Numerical strategies for quantum tomography: alternatives to full optimization,” Phys. Rev. A 79, 022109 (2009).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Jim’enez, O.

C. Paiva, E. Burgos-Inostroza, O. Jim’enez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A 82, 032115 (2010).
[CrossRef]

Jozsa, R.

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
[CrossRef]

Kadoya, Y.

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
[CrossRef]

Kasai, K.

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
[CrossRef]

Kaszlikowski, D.

T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[CrossRef]

D. Kaszlikowski, P. Gnaci’nski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubit,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[CrossRef] [PubMed]

Kaznady, M. S.

M. S. Kaznady, and D. F. V. James, “Numerical strategies for quantum tomography: alternatives to full optimization,” Phys. Rev. A 79, 022109 (2009).
[CrossRef]

Khan, I. A.

M. N. O. Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).

Klimov, A. B.

I. Sainz, L. Roa, and A. B. Klimov, “Unbiased nonorthogonal bases for tomographic reconstruction,” Phys. Rev. A 81, 052114 (2010).
[CrossRef]

A. B. Klimov, C. Muoz, A. Fern’andez, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303 (2008).
[CrossRef]

J. L. Romero, G. Bork, A. B. Klimov, and L. L. S’anchez-Soto, “Structure of the sets of mutually unbiased bases for N qubits,” Phys. Rev. A 72, 062310 (2005).
[CrossRef]

Kwek, L. C.

T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[CrossRef]

Kwiat, P. G.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

La Mela, C.

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Lamelle, D. S.

S. P. Walborn, D. S. Lamelle, M. P. Almeida, and P. H. Souto Ribeiro, “Quantum key distribution with higherorder alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[CrossRef] [PubMed]

Langford, N. K.

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).
[CrossRef]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Ledesma, S.

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Lima, G.

G. Lima, E. S. G’omez, A. Vargas, R. O. Vianna, and C. Saavedra, “Fast entanglement detection for unknown states of two spatial qutrits,” Phys. Rev. A 82, 012302 (2010).
[CrossRef]

G. Lima, A. Vargas, L. Neves, R. Guzm’an, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express 17, 10688–10696 (2009).
[CrossRef] [PubMed]

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. P’adua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre G’omez, C. Saavedra, and S. P’adua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
[CrossRef]

L. Neves, G. Lima, J. G. A. G’omez, C. H. Monken, C. Saavedra, and S. P’adua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef] [PubMed]

Maciel, T. O.

T. O. Maciel, and R. O. Vianna, “Viable entanglement detection of unknown mixed states in low dimensions,” Phys. Rev. A 80, 032325 (2009).
[CrossRef]

Miklaszewski, W.

D. Kaszlikowski, P. Gnaci’nski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubit,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[CrossRef] [PubMed]

Monken, C. H.

L. Neves, G. Lima, J. G. A. G’omez, C. H. Monken, C. Saavedra, and S. P’adua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef] [PubMed]

Munro, W. J.

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Muoz, C.

A. B. Klimov, C. Muoz, A. Fern’andez, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303 (2008).
[CrossRef]

Nemoto, K.

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[CrossRef]

Neves, L.

G. Lima, A. Vargas, L. Neves, R. Guzm’an, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express 17, 10688–10696 (2009).
[CrossRef] [PubMed]

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. P’adua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre G’omez, C. Saavedra, and S. P’adua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
[CrossRef]

L. Neves, G. Lima, J. G. A. G’omez, C. H. Monken, C. Saavedra, and S. P’adua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef] [PubMed]

O’Brien, J. L.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

P’adua, S.

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. P’adua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre G’omez, C. Saavedra, and S. P’adua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
[CrossRef]

L. Neves, G. Lima, J. G. A. G’omez, C. H. Monken, C. Saavedra, and S. P’adua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef] [PubMed]

Paiva, C.

C. Paiva, E. Burgos-Inostroza, O. Jim’enez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A 82, 032115 (2010).
[CrossRef]

Paz, J. P.

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Pryde, G. J.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Puentes, G.

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Renes, J. M.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2181 (2004).
[CrossRef]

Roa, L.

I. Sainz, L. Roa, and A. B. Klimov, “Unbiased nonorthogonal bases for tomographic reconstruction,” Phys. Rev. A 81, 052114 (2010).
[CrossRef]

Romero, J. L.

J. L. Romero, G. Bork, A. B. Klimov, and L. L. S’anchez-Soto, “Structure of the sets of mutually unbiased bases for N qubits,” Phys. Rev. A 72, 062310 (2005).
[CrossRef]

Roos, C. F.

H. Haffner, W. Hansel, C. F. Roos, and J. Benhelm, “D. Chek-al-kar, M. Chwalla, T. Korber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne,W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[CrossRef] [PubMed]

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

S’anchez-Soto, L. L.

J. L. Romero, G. Bork, A. B. Klimov, and L. L. S’anchez-Soto, “Structure of the sets of mutually unbiased bases for N qubits,” Phys. Rev. A 72, 062310 (2005).
[CrossRef]

Saavedra, C.

G. Lima, E. S. G’omez, A. Vargas, R. O. Vianna, and C. Saavedra, “Fast entanglement detection for unknown states of two spatial qutrits,” Phys. Rev. A 82, 012302 (2010).
[CrossRef]

G. Lima, A. Vargas, L. Neves, R. Guzm’an, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express 17, 10688–10696 (2009).
[CrossRef] [PubMed]

A. B. Klimov, C. Muoz, A. Fern’andez, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303 (2008).
[CrossRef]

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. P’adua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre G’omez, C. Saavedra, and S. P’adua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
[CrossRef]

L. Neves, G. Lima, J. G. A. G’omez, C. H. Monken, C. Saavedra, and S. P’adua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef] [PubMed]

Sainz, I.

I. Sainz, L. Roa, and A. B. Klimov, “Unbiased nonorthogonal bases for tomographic reconstruction,” Phys. Rev. A 81, 052114 (2010).
[CrossRef]

Santos, I. F.

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre G’omez, C. Saavedra, and S. P’adua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
[CrossRef]

Saraceno, M.

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

Scott, A. J.

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2181 (2004).
[CrossRef]

Souto Ribeiro, P. H.

S. P. Walborn, D. S. Lamelle, M. P. Almeida, and P. H. Souto Ribeiro, “Quantum key distribution with higherorder alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[CrossRef] [PubMed]

Spreeuw, R. J. C.

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[CrossRef]

Steinberg, A. M.

R. B. A. Adamson, and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
[CrossRef] [PubMed]

Taguchi, G.

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
[CrossRef]

Thew, R. T.

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[CrossRef]

Torres-Ruiz, F. A.

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. P’adua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

Vargas, A.

G. Lima, E. S. G’omez, A. Vargas, R. O. Vianna, and C. Saavedra, “Fast entanglement detection for unknown states of two spatial qutrits,” Phys. Rev. A 82, 012302 (2010).
[CrossRef]

G. Lima, A. Vargas, L. Neves, R. Guzm’an, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express 17, 10688–10696 (2009).
[CrossRef] [PubMed]

Vianna, R. O.

G. Lima, E. S. G’omez, A. Vargas, R. O. Vianna, and C. Saavedra, “Fast entanglement detection for unknown states of two spatial qutrits,” Phys. Rev. A 82, 012302 (2010).
[CrossRef]

T. O. Maciel, and R. O. Vianna, “Viable entanglement detection of unknown mixed states in low dimensions,” Phys. Rev. A 80, 032325 (2009).
[CrossRef]

Walborn, S. P.

S. P. Walborn, D. S. Lamelle, M. P. Almeida, and P. H. Souto Ribeiro, “Quantum key distribution with higherorder alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[CrossRef] [PubMed]

White, A. G.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

Wootters, W. K.

W. K. Wootters, and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989).
[CrossRef]

Yoshimoto, N.

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
[CrossRef]

Zeilinger, A.

D. Kaszlikowski, P. Gnaci’nski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubit,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[CrossRef] [PubMed]

Zukowski, M.

D. Kaszlikowski, P. Gnaci’nski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubit,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[CrossRef] [PubMed]

Ann. Phys. (1)

W. K. Wootters, and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989).
[CrossRef]

J. Math. Phys. (1)

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. 45, 2171–2181 (2004).
[CrossRef]

J. Mod. Opt. (1)

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994).
[CrossRef]

J. Phys. A: Math. Theor. (1)

I. D. Ivanovic, “Geometrical description of quantal state determination,” J. Phys. A: Math. Theor. 14, 3241–3245 (1981).
[CrossRef]

J. Phys. B (1)

G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. P’adua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

Nature (1)

H. Haffner, W. Hansel, C. F. Roos, and J. Benhelm, “D. Chek-al-kar, M. Chwalla, T. Korber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Guhne,W. Dur, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature 438, 643–646 (2005).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Phys. Rev. A (15)

T. O. Maciel, and R. O. Vianna, “Viable entanglement detection of unknown mixed states in low dimensions,” Phys. Rev. A 80, 032325 (2009).
[CrossRef]

G. Lima, E. S. G’omez, A. Vargas, R. O. Vianna, and C. Saavedra, “Fast entanglement detection for unknown states of two spatial qutrits,” Phys. Rev. A 82, 012302 (2010).
[CrossRef]

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[CrossRef]

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre G’omez, C. Saavedra, and S. P’adua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
[CrossRef]

A. B. Klimov, C. Muoz, A. Fern’andez, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303 (2008).
[CrossRef]

T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[CrossRef]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[CrossRef]

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).
[CrossRef]

I. Sainz, L. Roa, and A. B. Klimov, “Unbiased nonorthogonal bases for tomographic reconstruction,” Phys. Rev. A 81, 052114 (2010).
[CrossRef]

C. Paiva, E. Burgos-Inostroza, O. Jim’enez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A 82, 032115 (2010).
[CrossRef]

G. Taguchi, T. Dougakiuchi, N. Yoshimoto, K. Kasai, M. Iinuma, H. F. Hofmann, and Y. Kadoya, “Measurement and control of spatial qubits generated by passing photons through double slits,” Phys. Rev. A 78, 012307 (2008).
[CrossRef]

M. S. Kaznady, and D. F. V. James, “Numerical strategies for quantum tomography: alternatives to full optimization,” Phys. Rev. A 79, 022109 (2009).
[CrossRef]

G. Puentes, C. La Mela, S. Ledesma, C. Iemmi, J. P. Paz, and M. Saraceno, “Optical simulation of quantum algorithms using programmable liquid-crystal displays,” Phys. Rev. A 69, 042319 (2004).
[CrossRef]

J. L. Romero, G. Bork, A. B. Klimov, and L. L. S’anchez-Soto, “Structure of the sets of mutually unbiased bases for N qubits,” Phys. Rev. A 72, 062310 (2005).
[CrossRef]

Phys. Rev. Lett. (6)

R. B. A. Adamson, and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
[CrossRef] [PubMed]

D. Kaszlikowski, P. Gnaci’nski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled n-dimensional systems are stronger than for two qubit,” Phys. Rev. Lett. 85, 4418–4421 (2000).
[CrossRef] [PubMed]

L. Neves, G. Lima, J. G. A. G’omez, C. H. Monken, C. Saavedra, and S. P’adua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005).
[CrossRef] [PubMed]

M. N. O. Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005).

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

S. P. Walborn, D. S. Lamelle, M. P. Almeida, and P. H. Souto Ribeiro, “Quantum key distribution with higherorder alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957).
[CrossRef]

Other (1)

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Photonic State Tomography, Advances in AMO Physics (Elsevier, 2006), Vol. 52, Chap. 3.

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

Experimental Setup. See the main text for details.

Fig. 2
Fig. 2

The initial qudit-7 state. In (a) one can see the aperture modulated at the first SLM. In (b) there is a comparison between the laser beam spatial profile and this aperture.

Fig. 3
Fig. 3

Phases modulated at the second SLM and the corresponding recorded interference patterns. Figure (a) illustrates the relation between the gray level of the second LCD and the phase being modulated by this modulator. Some of the modulations used in the MUB-QT, for projecting the generated qudit-7 state onto | ψ m ( α ) , are shown in the insets of figures (b), (c) and (d). In (b) α = 7 and m = 1. In (c) α = 7 and m = 2, and in (d) α = 7 and m = 5. The corresponding recorded interference patterns (points) are compared with the expected ones (lines). The expected patterns are calculated from the amplitudes of |Ψ7expc. The integration time of these measurements was one second and the maximal single count rate recorded was 10000 counts/s.

Fig. 4
Fig. 4

MUB-QT of the generated qudit-7 state. (a) The expected probabilities based on the predicted state |Ψ7expc. (b) The recorded probabilities with single counts. (c) The real and the imaginary parts of the reconstructed state. On the insets of (c) the parts of the expected density operator are shown.

Fig. 5
Fig. 5

Generation and reconstruction of the spatial qudit-8 states. In (a) [(d)] there is a comparison between the object being modulated at the first SLM and the spatial laser profile used for generating the first (second) qudit-8 state. In (b) and (e) one can see a comparison between the recorded probabilities and the expected ones (insets) that are calculated from the states | Ψ 8 ( 1 ) expc and | Ψ 8 ( 2 ) expc, respectively. In (c) [(f)] the corresponding reconstructed density operator for the first (second) qudit-8 state generated is shown. On the insets of (c) and (f) one can see the expected density operators.

Equations (18)

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

ρ = α = 1 D + 1 m = 1 D p m ( α ) m ( α ) I ,
| Ψ = l = l D l D β l | l ,
| l a π dq e iqld sinc ( q a ) | 1 q .
| Ψ mod 1 = 1 N l = l D l D λ l β l | l ,
| Ψ mod 2 = 1 N l = l D l D λ l β l e i θ l | l ,
C ( x ) | vac | E s ( + ) ( x , z m ) | ψ mod 2 | 2 sin c 2 ( k x a f 3 ) | l β l λ l e i θ l e i l d k x f 3 | 2
C ( 0 ) | l β l λ l e i θ l | 2 .
| ψ m ( α ) = l ε m , l ( α ) e i φ m , l ( α ) | l .
ε m , l ( α ) = 1 7 , φ m , l ( α ) = 2 π ( α l 2 + m l ) 7 ,
U ( 1 ) = 1 8 ( 1 i 1 i 1 i 1 i i 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 i i 1 i 1 i 1 i 1 i 1 i 1 i 1 i 1 1 i 1 i 1 i 1 i ) .
U ( 2 ) = 1 8 ( 1 1 1 1 i i i i 1 1 1 1 i i i i 1 1 1 1 i i i i 1 1 1 1 i i i i i i i i 1 1 1 1 i i i i 1 1 1 1 i i i i 1 1 1 1 i i i i 1 1 1 1 ) .
U ( 3 ) = 1 2 ( 1 0 1 0 0 0 0 0 0 i 0 i 0 0 0 0 0 1 0 1 0 0 0 0 i 0 i 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 i 0 i 0 0 0 0 0 1 0 1 0 0 0 0 i 0 i 0 ) .
U ( 4 ) = 1 2 ( 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 ) .
U ( 5 ) = 1 8 ( 1 i i 1 1 i i 1 i 1 1 i i 1 1 i i 1 1 i i 1 1 i 1 i i 1 1 i i 1 i 1 1 i i 1 1 i 1 i i 1 1 i i 1 1 i i 1 1 i i 1 i 1 1 i i 1 1 i ) .
U ( 6 ) = 1 2 ( 1 0 1 0 1 0 1 0 0 i 0 i 0 i 0 i 1 0 1 0 1 0 1 0 0 i 0 i 0 i 0 i 0 1 0 1 0 1 0 1 i 0 i 0 i 0 i 0 0 1 0 1 0 1 0 1 i 0 i 0 i 0 i 0 ) .
U ( 7 ) = 1 2 ( 1 i 0 0 1 i 0 0 i 1 0 0 i 1 0 0 0 0 1 i 0 0 1 i 0 0 i 1 0 0 i 1 0 0 i 1 0 0 i 1 0 0 1 i 0 0 1 i i 1 0 0 i 1 0 0 1 i 0 0 1 i 0 0 ) .
U ( 8 ) = 1 8 ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ) .
U ( 9 ) = 1 2 ( 1 0 i 0 1 0 i 0 0 1 0 i 0 1 0 i i 0 1 0 i 0 1 0 0 i 0 1 0 i 0 1 i 0 1 0 i 0 1 0 0 i 0 1 0 i 0 1 1 0 i 0 1 0 i 0 0 1 0 i 0 1 0 i ) .

Metrics