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 OSA

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  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, 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.
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    [CrossRef]
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    [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. Gnaciński, 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. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “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. Guzmán, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express 17, 10688–10696 (2009).
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  18. G. Lima, L. Neves, I. F. Santos, J. G. Aguirre Gómez, C. Saavedra, and S. Pádua, “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. Pádua, “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 wuantum-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. Gómez, 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 higher-order 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. Sánchez-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. Jiménez, 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. Gómez, 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. Jiménez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A 82, 032115 (2010).
[CrossRef]

2009 (3)

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]

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

2008 (4)

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

G. Lima, F. A. Torres-Ruiz, L. Neves, A Delgado, C Saavedra, and S. Pádua, “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)

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

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

2005 (4)

L. Neves, G. Lima, J. G. A. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “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, 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]

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

2004 (4)

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]

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]

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. Gnaciński, 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 wuantum-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 Gómez, J. G.

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre Gómez, C. Saavedra, and S. Pádua, “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 higher-order alphabets using spatially encoded qudits,” Phys. Rev. Lett. 96, 090501 (2006).
[CrossRef] [PubMed]

Altepeter, J. B.

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

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]

Becher, C.

H. Haffner, W. Hansel, C. F. Roos, 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]

Benhelm, J.

H. Haffner, W. Hansel, C. F. Roos, 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]

Blatt, R.

H. Haffner, W. Hansel, C. F. Roos, 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. Sánchez-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. Jiménez, 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]

Chek-al-kar, D.

H. Haffner, W. Hansel, C. F. Roos, 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]

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]

Chwalla, M.

H. Haffner, W. Hansel, C. F. Roos, 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]

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

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

Delgado, A.

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

Dur, W.

H. Haffner, W. Hansel, C. F. Roos, 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]

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 wuantum-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]

Fernández, A.

A. B. Klimov, C. Muoz, A. Fernández, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303(R) (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]

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]

Gnacinski, P.

D. Kaszlikowski, P. Gnaciński, 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]

Gómez, E. S.

G. Lima, E. S. Gómez, 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]

Gómez, J. G. A.

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

Guhne, O.

H. Haffner, W. Hansel, C. F. Roos, 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]

Guzmán, R.

Haffner, H.

H. Haffner, W. Hansel, C. F. Roos, 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]

Hansel, W.

H. Haffner, W. Hansel, C. F. Roos, 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]

Jeffrey, E. R.

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

Jiménez, O.

C. Paiva, E. Burgos-Inostroza, O. Jiménez, 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. Gnaciński, 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. Fernández, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303(R) (2008).
[CrossRef]

Klimov, A.B.

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

Korber, T.

H. Haffner, W. Hansel, C. F. Roos, 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]

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]

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

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 higher-order 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. Gómez, 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. Guzmán, 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. Pádua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

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

L. Neves, G. Lima, J. G. A. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “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. Gnaciński, 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. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “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. Fernández, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303(R) (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. Guzmán, 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. Pádua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

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

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

O.-Hale, M. N.

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).

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]

Pádua, S.

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

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

L. Neves, G. Lima, J. G. A. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “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. Jiménez, 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 wuantum-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]

Rapol, U. D.

H. Haffner, W. Hansel, C. F. Roos, 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]

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]

Riebe, M.

H. Haffner, W. Hansel, C. F. Roos, 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]

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. Sánchez-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, 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 wuantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[CrossRef]

Saavedra, C

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

Saavedra, C.

G. Lima, E. S. Gómez, 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. Guzmán, 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. Fernández, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303(R) (2008).
[CrossRef]

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

L. Neves, G. Lima, J. G. A. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “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]

Sánchez-Soto, L.L.

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

Santos, I. F.

G. Lima, L. Neves, I. F. Santos, J. G. Aguirre Gómez, C. Saavedra, and S. Pádua, “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]

Schmidt, P. O.

H. Haffner, W. Hansel, C. F. Roos, 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]

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 higher-order 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. Pádua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

Vargas, A.

G. Lima, E. S. Gómez, 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. Guzmán, 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. Gómez, 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 higher-order 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. Gnaciński, 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. Gnaciński, 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. Pádua, “Measurement of spatial qubits,” J. Phys. B 41, 185501 (2008).
[CrossRef]

Nature (1)

H. Haffner, W. Hansel, C. F. Roos, 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 wuantum-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. Gómez, 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]

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. Jiménez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A 82, 032115 (2010).
[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. Sánchez-Soto, “Structure of the sets of mutually unbiased bases for N qubits,” Phys. Rev. A 72, 062310 (2005).
[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. Lima, L. Neves, I. F. Santos, J. G. Aguirre Gómez, C. Saavedra, and S. Pádua, “Propagation of spatially entangled qudits through free space,” Phys. Rev. A 73, 032340 (2006).
[CrossRef]

A. B. Klimov, C. Muoz, A. Fernández, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A 77, 060303(R) (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]

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. Gnaciński, 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. Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “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).

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Rev. Mod. Phys. (1)

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Other (1)

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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)

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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 ) .

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