## Experimental quantum tomography of photonic qudits via mutually unbiased basis |

Optics Express, Vol. 19, Issue 4, pp. 3542-3552 (2011)

http://dx.doi.org/10.1364/OE.19.003542

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### 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

## 1. Introduction

*D*-dimensional quantum system (qudit-

*D*) is represented by a positive semidefinite, unit-trace Hermitian operator, which requires only

*D*

^{2}− 1 independent real numbers for its specification. In the standard QT [1

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]

*D*= 2

*(i.e., a composite system of*

^{N}*N*qubits), these parameters are determined by projecting the density operator onto completely factorized bases [2

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]

*N*qubits, for example, it is necessary to perform 6

*projections [9]. In the case of using MUB-QT, the number of measurements adopted is minimal [5*

^{N}5. W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys **191**, 363–381 (1989). [CrossRef]

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

*N*qubits, it is necessary to perform only 2

*(2*

^{N}*+ 1) projections.*

^{N}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]

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). [CrossRef] [PubMed]

*F*) [17

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

*ψ*〉) and the reconstructed one (

*ρ*),

*F*≡ 〈

*ψ*|

*ρ*|

*ψ*〉. For the 7- and 8-dimensional reconstructed qudit states, the fidelities obtained were greater than 90%.

## 2. Setup description

*D*-dimensional Hilbert space, whose dimension is a prime or a prime power number, there exist

*D*+ 1 MUBs. The constructive procedures for obtaining them have been explicitly given in [5

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

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

*D*is represented in terms of the MUBs by [10

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

*ρ*onto the

*m*-th state

*α*MUB, with

*α*= 1, 2, ...,

*D*+ 1. The MUB-QT is performed by determining the

*D*-multi slit as shown in Fig. 1. The slit’s width is 2

*a*= 104

*μ*m and the distance between two consecutive slits is

*d*= 208

*μ*m. The state of the single photon transmitted through the SLM can be written as [14

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]

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). [CrossRef] [PubMed]

*l*= (

_{D}*D*− 1)/2 and the state |

*l*〉 is a single photon state defined, up to a global phase factor, as It represents the state of the photon transmitted by the

*l*th-slit of the SLM. |1

*q*〉 is the Fock state of a photon with the transverse wave vector

*q*⃗. The |

*l*〉 states form the logical basis in the

*D*-dimensional Hilbert space of the transmitted photons, whose dimension

*D*is defined by the number of slits addressed in the SLM. In this work we considered the generation of 7- and 8-dimensional spatial qudit states. Higher dimensional qudits can, in principle, also be generated by considering more complexes apertures. The

*β*coefficients are dependent on the spatial profile and the wavefront curvature of the laser beam at the SLM plane. The first SLM can also be used to control each slit transmission independently, such that the initial state of Eq. (2) can be modified to where

_{l}*t*representing the transmission of the

_{l}*l*-slit, and

*N*is the normalization constant given by

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). [CrossRef] [PubMed]

*f*-configuration, such that the LCD of the second modulator is at a distance of 30 cm from the first LCD. This SLM is used to modulate the phase of the beam, independently, at each point of slit image formation. At the plane of image formation, the diffracted photons can again be described as spatial qudit states [18

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]

_{mod}_{1}can then be modified to where

*θ*is the phase given at the image of the

_{l}*l*-th slit. Therefore, the two SLMs allow for a full manipulation of the initial spatial qudit states.

_{2}, such that L

_{1}and L

_{2}form a telescope with a magnification factor of 3.3. The collimated beam is then focused by a 1 m focal length lens (

*L*

_{3}), and a point-like detector (APD) is used to record the single photons at the center of the interference pattern formed in the Fourier transform plane. The CCD camera, shown in Fig. 1, was used just for the initial alignment of the SLMs without attenuating the laser beam. The point-like detector is composed of a conventional avalanche photo-counting module with a slit of 20

*μ*m in front of it. The spatial filtering being introduced by the point-like detector is the last ingredient necessary for projecting the spatial qudit states onto the MUBs’ vectors [19

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]

*x*at the focal plane of lens

*L*

_{3}(

*z*−plane) is given by: where we have used that

_{m}*E*

^{(+)}(

*x*,

*z*) ∝

_{m}*â*

_{kx/f3}and

*f*

_{3}is the focal length of lens

*L*

_{3}. In our case, when the point-like detector is fixed at the center of the interference pattern formed, the single count rate will be

*m*-th spatial vector of the α-th MUB is given by

*m*-th projector of the

*α*-th MUB. This is done by changing the gray level of the apertures modulated in the first and the second LCD(s), respectively. For an arbitrary initial quantum state

*ρ*, using the linear properties of quantum theory, we have that

*D*is a prime number, all the probabilities for a given MUB, can also be obtained from a single interference pattern formed by a specific phase modulation of the second SLM. In this case, all the probabilities of the MUB considered can be inferred by measuring the single counts along the transversal direction

*x*. This means that with a single experimental configuration one can get,

*only in this specific case*, all the projective measurements of a given standard MUB labeled by the index

*α*.

## 3. Experimental results

### 3.1. qudit-7 state preparation

*β*are determined by the spatial distribution of the beam and by the aperture addressed in the SLM1. The apperture modulated in this SLM is shown in Fig. 2(a). In Fig. 2(b) there is a comparison between the laser beam spatial profile and this aperture. For such configuration, we can expect that the spatial qudit state being generated in the experiment is described by |Ψ

_{l}_{7}〉

*= 0.256|−3〉 + 0 .362|−2〉 + 0.443|−1〉 + 0.473|0〉 + 0.439|1〉 + 0.352|2〉 + 0.254|3〉.*

_{expc}### 3.2. qudit-7 state reconstruction

**191**, 363–381 (1989). [CrossRef]

*α*= 1, 2, ..., 7 and

*m*,

*π*). The probabilities for projecting onto the logical base vectors, which form one of the MUBs for a 7-dimensional Hilbert space, can be determined by measuring the single counts at the plane of image formation. This is done by removing the second SLM and scanning transversely the APD at this plane [18

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]

_{7}〉

*onto some of the vectors of the 7-th MUB. The corresponding interference patterns were completely recorded just to show the purity of the modified states. The visibility of these patterns guarantees that the modulation is not introducing decoherence to the qudit states, showing that the modified states are nearly pure [16*

_{expc}**17**, 10688–10696 (2009). [CrossRef] [PubMed]

_{7}〉

*. One can clearly see that the obtained probabilities are very close to the expected ones.*

_{expc}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]

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]

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

*F*

_{7}

*= 0.96 ± 0.03. This error was calculated taking into account the Poissonian distribution for the single counts recorded in the experiment. The errors on the elements of the reconstructed state can be calculated directly from Eq. (1). Experimental obtained fidelities are usually limited by the imperfections of an experimental setup. We believe that in our case, these imperfections are mainly due to the mismatch that can occur while superposing the image of the first SLM onto the image modulated on the LCD of the second SLM.*

_{q}### 3.3. Preparing the qudit-8 states

*l*〉 states as follows:

*dimensional qudit, with*

^{N}*N*integer, this experimental setup allows us for mimicking a multipartite system of

*N*qubits. These subsystems are encoded on the transverse modes of the transmitted photon and, therefore, they cannot be used to test the quantum non-locality [22

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]

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]

### 3.4. Reconstruction of the qudit-8 states

11. 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]

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]

**77**, 060303(R) (2008). [CrossRef]

*α*= 1, 2, ..., 8 and

*m*,

*l*= −7/2, −5/2, ..., 7/2. For generating these spatial projections it was necessary to change the transmission of the slits addressed at the first SLM.

## 4. Conclusion

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]

## A. Unitary transformations used for the qudit-8 MUB-QT

*l*〉 of Eq. (2), which define the new family of MUBs used for the qudit-8 MUB-QT [11

**77**, 060303(R) (2008). [CrossRef]

*α*in

*U*(

*α*) denotes a specific mutually unbiased basis in the 8-dimensional Hilbert space. It can be noted that in transformations with

*α*= 1, 2, 5, 8, all the

*α*= 4, 6, 7, 8 we get that

*α*= 3 that

_{1}. Furthermore, the values for the phase modulation at the SLM

_{2}[Eq. (8)], are given by

*π*/2,

*π,*3

*π*/2}. Hence the LCD must allow for full modulation at the {0,2

*π*} domain, which is effectively provided by the used SLM as it is shown in Fig. 3(a).

## Acknowledgments

## References and links

1. | U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. |

2. | D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A |

3. | R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A |

4. | M. D. de Burgh, N. K. Langford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,” Phys. Rev. A |

5. | W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys |

6. | R. B. A. Adamson and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. |

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 |

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. |

9. | J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, |

10. | I. D. Ivanovic, “Geometrical description of quantal state determination,” J. Phys. A: Math. Theor. |

11. | A. B. Klimov, C. Muoz, A. Fernández, and C. Saavedra, “Optimal quantum-state reconstruction for cold trapped ions,” Phys. Rev. A |

12. | T. Durt, D. Kaszlikowski, J. L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A |

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. |

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. |

15. | M. N. O.-Hale, I. A. Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled |

16. | G. Lima, A. Vargas, L. Neves, R. Guzmán, and C. Saavedra, “Manipulating spatial qudit states with programmable optical devices,” Opt. Express |

17. | R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. |

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 |

19. | G. Lima, F. A. Torres-Ruiz, L. Neves, A Delgado, C Saavedra, and S. Pádua, “Measurement of spatial qubits,” J. Phys. B |

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 |

21. | M. S. Kaznady and D. F. V. James, “Numerical strategies for quantum tomography: alternatives to full optimization,” Phys. Rev. A |

22. | A. Einstein, B. Podolsky, and N. Rosen, “Can wuantum-mechanical description of physical reality be considered complete?” Phys. Rev. |

23. | T. O. Maciel and R. O. Vianna, “Viable entanglement detection of unknown mixed states in low dimensions,” Phys. Rev. A |

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 |

25. | R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A |

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. |

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 |

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 |

29. | J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” J. Math. Phys. |

30. | I. Sainz, L. Roa, and A. B. Klimov, “Unbiased nonorthogonal bases for tomographic reconstruction,” Phys. Rev. A |

31. | C. Paiva, E. Burgos-Inostroza, O. Jiménez, and A. Delgado, “Quantum tomography via equidistant states,” Phys. Rev. A |

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: January 10, 2011

Revised Manuscript: January 28, 2011

Manuscript Accepted: January 30, 2011

Published: February 8, 2011

**Citation**

G. Lima, L. Neves, R. Guzmán, E. S. Gómez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, "Experimental quantum tomography of photonic qudits via mutually unbiased basis," Opt. Express **19**, 3542-3552 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3542

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### References

- U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys. 29, 74–93 (1957). [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]
- W. K. Wootters, and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys. 191, 363–381 (1989). [CrossRef]
- R. B. A. Adamson, and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, Photonic State Tomography, Advances in AMO Physics (Elsevier, 2006), Vol. 52, Chap. 3.
- I. D. Ivanovic, “Geometrical description of quantal state determination,” J. Phys. A: Math. Theor. 14, 3241–3245 (1981). [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. 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).
- 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]
- R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41, 2315–2323 (1994). [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]
- 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]
- M. S. Kaznady, and D. F. V. James, “Numerical strategies for quantum tomography: alternatives to full optimization,” Phys. Rev. A 79, 022109 (2009). [CrossRef]
- A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935). [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]
- 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]
- 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]
- 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]
- 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]
- 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]

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