We present a new method for preparing multidimensional spatial qudits by means of a single phase-only spatial light modulator (SLM). This method improves previous ones that use two SLMs, one working in amplitude regime and the other in phase regime. To that end, we addressed diffraction gratings on the slits that define the state and then we performed a spatial filtering in the Fourier plane. The amplitude of the coefficients of the quantum state are determined by the modulation deep of the diffraction gratings, and the relative phase is the mean phase value of the diffraction gratings. This encoding result to be more compact, less expensive and use the photons more efficiently.Instituto de Física La Plat

Arias, A.

Iemmi, C.

Ledesma, S.

Neves, L.

Rebón, Lorena

Solís-Prosser, A.M.A.

Varga, J.J.M.

Journal of Physics Conference Series

English

SEDICI - Repositorio de la UNLP

23rd Congress of the International Commission for Optics (ICO 23)
Journal of Physics: Conference Series 605 (2015) 012035
IOP Publishing
doi:10.1088/1742-6596/605/l/012035
Preparing arbitrary pure states of spatial qudits with 
a single phase-only spatial light modulator
J J M Varga x*, M A Solis-Prosser2, L. Rebón3, A Arias4, L
Neves5, C Iemmi1, S Ledesma1 
departamento de Física, FCEyN, Universidad de Buenos Aires, Buenos Aires 1428, Argentina 
3Center for Optics and Photonics, Universidad de Concepción, Casilla 4016, Concepción, Chile 
“Instituto de Física de La Plata, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, 
Argentina
venter for Research, Instituto Tecnológico Metropolitano, A.A 54954, Medellin, Colombia 
6Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG 
30123-970, Brazil
E-mail: miguelQdf .uba. ar
Abstract. We present a new method for preparing multidimensional spatial qudits by means 
of a single phase-only spatial light modulator (SLM). This method improves previous ones that 
use two SLMs, one working in amplitude regime and the other in phase regime. To that end, 
we addressed diffraction gratings on the slits that define the state and then we performed a 
spatial filtering in the Fourier plane. The amplitude of the coefficients of the quantum state 
are determined by the modulation deep of the diffraction gratings, and the relative phase is the 
mean phase value of the diffraction gratings. This encoding result to be more compact, less 
expensive and use the photons more efficiently.
Quantum systems are the information carriers in quantum information processing and 
computing protocols. While qubits (two-level systems) are the usual and most basic systems 
to carry out such tasks, qudits [D-level systems (£>>2)] have been attracting growing interest 
due to their greater potential for those applications [1]. In particular, single photons are the 
natural choice for communications since they are easily transportable and have several degrees of 
freedom (DOFs) to encode information. In the case of using the transverse momentum-position 
encoding [2, 3], the simplest approach is achieved by discretizing the one-dimensional transverse 
modes of the photons when they are made to pass through an aperture with D slits which sets 
the dimension of these so-called spatial qudits.
Spatial qudits are relatively simple to generate and offer the possibility of working in high 
dimensions without cumbersome optical setups. Recently, this encoding has drawn interest for 
miscellaneous applications such as quantum information protocols [4], quantum games [5], and 
quantum key distribution [6]. Therefore, the ability to prepare arbitrary pure states of such 
systems represents an important step toward realizations of quantum optics experiments based 
on it. Static amplitude and phase masks could be used for this purpose, but in practice it would 
be extremely difficult and time-consuming as each state intended to be prepared would require 
to set up the corresponding masks in the apparatus. Programmable spatial light modulators 
(SLMs) can dramatically simplify this process, allowing real-time manipulations of the state 
coefficients without physically aligning any optical components. In this regard, it has been
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution 
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd 1
23rd Congress of the International Commission for Optics (ICO 23) IOP Publishing
doi:10.1088/1742-6596/605/l/012035
preparation
Polarizer
Spatial filter/ 
beam expander
Diode Laser
@647 nm
ND 
filters
Journal of Physics: Conference Series 605 (2015) 012035
(Image plane)
(phase)
Beam dumpcamera
State characterization (Fourier plane) CCD camera
Figure 1. Experimental setup
shown [7] that by imaging the output beam of an amplitude-only SLM onto a phase-only SLM 
allows one to get complete and independent control of the amplitude and phase of the complex 
coefficients that define the qudit state. However, the use of two SLMs, besides being more 
expensive, entails two drawbacks: (i) the overall diffraction efficiency is very low, and (ii) in 
order to avoid even more losses, the image of the first SLM must match the second one pixel by 
pixel, which is difficult in practice.
Here, we present a demonstration of a method for preparing arbitrary pure states of spatial 
qudits with a single phase-only SLM [8], which has a much higher diffraction efficiency and does 
not require imaging systems. Among the alternative methods to represent a complex function 
in a single SLM [9, 10] we choose, for simplicity, a technique where the amplitude information 
is encoded in a phase-only filter [11, 12], On one hand, the required amplitude is achieved 
by programming a phase grating, with an appropriate modulation into the slit regions. On 
the other hand, the required phase value is obtained by adding a constant phase value to the 
phase grating. Then, the first diffraction order is selected at the Fourier plane and the filtered 
information is antitransformed on the final plane (Fig. 1). The result is a light distribution 
corresponding to an image of the slits containing the complete complex modulation. Regarding 
the type of selected grating, either binary or blazed, the overall quality of the prepared states 
is very similar. However, blazed gratings provide better diffraction efficiency than binary ones. 
This enhanced efficiency can be important to increase the signal-to-noise ratio when working 
with single photon sources.
We have analyzed the performance of the proposed method by preparing a large number of 
arbitrary pure qubit states uniformly distributed on the surface of the Bloch sphere. To quantify 
the quality of the preparation we used the fidelity between the state intended to be prepared and 
the density matrix of the state actually prepared which is reconstructed by tomography (Fig. 2). 
The tomographic process is performed by measurements at different transverse positions in the 
near and in the far field [13]. In both cases, either binary or blazed grating, we obtained fidelities 
above 96% and average fidelities of 99.6%. We have found lor // 3 and D = 7 that the fidelities
were above 94%. All these relevant results are summerized in the Table 1.
We compared the luminous efficiency of our setup with the one that employed two SLMs [7]. 
In the case of using a blazed grating, our scheme is times more efficient, where r/ represents 
the light attenuation due to polarizing elements needed to obtain pure amplitude modulation
2
23rd Congress of the International Commission for Optics (ICO 23)
Journal of Physics: Conference Series 605 (2015) 012035
IOP Publishing
doi: 10.1088/1742-6596/605/1/012035
Table 1. Average fidelities of preparation for D-level qudits.
D 2 3 4 5 7
# of states/grating 561 24 70 25 94
-/bin 0.996 0.995 0.985 0.968 0.970
a>la 0.996 0.996 0.991 0.971 0.977
Figure 2. Ten-pixels period gratings: (a) binary and (b) blazed, (c) (Upper panels) amplitude 
and phase components of the aperture T(x) for preparing the state 0.56|0) +0.83ez0,637r 11). (Lower 
panels) corresponding phase masks for preparing this state at the first diffraction order, (d) 
Normalized intensity distributions and theoretical prediction for measurements of this state at 
near (upper panel) and far field (lower panel), (e) Real and imaginary parts of the experimentally 
reconstructed density matrix using the blazed grating. The ideal density matrix is shown as a 
blue wire grid and the obtained fidelity is F = 0.996.
with a twisted nematic SLM [15, 16]. The factor tj depends on the parameters involved in each 
particular case (wavelength, twist angle, birefringence of the SLM, etc.), but a typical value is 
about 0.1. Therefore, the method presented here appears as a valuable tool for experiments based 
on this encoding. In addition, it can be useful for assisting the measurement of high-dimensional 
photonic qudits, also including mixed states, enabling not only the storing of information but 
also the read out of it. It enables the implementation of the strategy developed in [14] by 
proyecting the image of the state to be measured at SLM and addresing at this device the 
phase grating function corresponding to the measurement state. A detection at the center of the 
filtered diffraction order at the Fourier plane will acomplish the process allowing one to obtain 
the statistics of arbitrary observables.
Finally, it is important to remark that although we have obtained a good quality of
3
23rd Congress of the International Commission for Optics (ICO 23)
Journal of Physics: Conference Series 605 (2015) 012035
IOP Publishing
doi:10.1088/1742-6596/605/l/012035
Table 2. Selected arbitrary states
state 1 |^i) = 0.70|0> I-0.70c7l-o<>7r|I)
state 2 ¡^2) = 0.70|0) + 0.70e*O37r|l)
state 3 ¡^3) = —0.69|0> + 073en-357r|l)
state 4 = —0.92|0> + 0.40eil’9O’r| 1>
state 5 = 0.48|0> + 0.88eiO-S77r11>
state 6 \^6) = -0.48|0) + 0.88en'557r|l)
time [ms]
1
0.99
0.98
13
TS 0.97
0.96
0.95
tIssr
Figure 3. Temporal phase fluctuation model for the SLM.
—state 2
---- state 3
---- state 4
---- state 5
—state 6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Phase shift between SLMs [rad/27r]
Figure 4. Fidelity vs. phase shift between SLMs, obtained by means of numerical simulation 
for six arbitrary states (D = 2) using the phase adition method.
preparation, the method presents an inherent phase fluctuation associated to the used SLMs, 
in our case liquid crystal on silicon displays (LCoS). We have selected LCoSs due to their high
4
23rd Congress of the International Commission for Optics (ICO 23) IOP Publishing
Journal of Physics: Conference Series 605 (2015) 012035 doi:10.1088/1742-6596/605/l/012035
spatial resolution and very high light efficiency. Nevertheless, these displays may lead to flicker in 
the optical beam because of their addressing scheme (pulse width modulation) which introduces 
temporal phase fluctuations [17, 18] that affect the quality of the encoded state. In order to 
study this effect we have analyzed the performance of the proposed method by simulating the 
preparation of some selected arbitrary states in one SLM and their tomographic reconstruction 
in a second SLM. For this purpose we have used a simplified model for the typical temporal 
fluctuations of the SLMs that consists on a triangular shaped fluctuation as is depicted in Fig. 3. 
Given that the optical configuration contain two similar SLMs that could be desynchronized, 
we have considered different phase shift between both modulators. Numerical simulations of the 
fidelity vs. the phase shift are shown in Fig. 4 for six arbitrary states. These states are shown 
in Table 2 As it can be noted fidelities are almost independent of the phase shift between the 
phase fluctuations in each SLM. This would indicate that it is not necessary to deal with the 
temporal synchronization of the modulators.
Conclusions
We have shown that a single phase-only SLM enables the preparation of arbitrary pure states 
of spatial qudits. Mean values of fidelities of preparation above 94% were obtained and a higher 
efficiency (at least 10 times) with respect to other methods is expected. It is important to note 
that this method can be useful not only for preparation but also for assisting the measurement of 
such systems. It enables the implementation of a strategy based on imaging the state represented 
on the first SLM onto the second SLM. In this case the first SLM is used to represent the state 
to be analized and the second one to represent the reconstruction basis. Therefore, the method 
presented here may become a valuable tool for experiments using spatial qudits.
Acknowledgment
We thank P. Kolenderski and A. Calatayud for helpful discussions. This work was supported by 
CONICYT PFB08-24 and Milenio ICM P10-030-F (Chile) and by UBACyT 20020100100689, 
CONICET PIP 112-200801-03047, and ANPCYT PICT 2010-02179 (Argentina).
References
[1] S. M. Barnett, Quantum Information (Oxford University Press, Oxford, 2009).
[2] L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled
states of qudits using twin photons”, Phys. Rev. Lett. 94, I0050I (2005).
[3] M. N. O’Sullivan-Hale, I. Ali-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).
[4] M. A. Solis-Prosser and L. Neves, “Remote state preparation of spatial qubits”, Phys. Rev. A 84, 012330
(2011).
[5] P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein,
“Aharon-Vaidman quantum game with a Young-type photonic qutrit”, Phys. Rev. A 86, 0I232I (2012).
[6] S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, “Quantum
key distribution session with 16-dimensional photonic states”, Sci. Rep. 3, 2316 (2013).
[7] 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 (2011).
[8] M. A. Solis-Prosser, A. Arias, J. J. M. Varga, L. Rebñ, S. Ledesma, C. Iemmi, and L. Neves, ’’Preparing
arbitrary pure states of spatial qudits with a single phase-only spatial light modulator” , Opt. Lett. 38, 4762 
(2013)
[9] R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image
and diffraction plane pictures”, Optik 35, 237 (1972).
[10] E. G. van Putten, I. M. Vellekoop, and A. P. Mosk, “Spatial amplitude and phase modulation using
commercial twisted nematic LCDs”, Appl. Opt. 47, 2076 (2008).
[11] J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto
phase-only filters”, Appl. Opt. 38, 5004 (1999).
5
IOP Publishing
doi:10.1088/1742-6596/605/1/012035
23rd Congress of the International Commission for Optics (ICO 23) 
Journal of Physics: Conference Series 605 (2015) 012035
[12] V. Bagnoud and J. D. Zuegel, “Independent phase and amplitude control of a laser beam by use of a single-
phase-only spatial light modulator”, Opt. Lett. 29, 295 (2004).
[13] G. Lima, F. A. Torres-Ruiz, L. Neves, A. Delgado, C. Saavedra, and S. Pdua, “Measurement of spatial
qubits”, J. Phys. B 41, 185501 (2008).
[14] 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 (2011).
[15] A. Márquez, C. Iemmi, I. Moreno, J. Davis, J. Campos, and M. Yzuel, “Quantitative predictions of the
modulation behavior of twisted nematic liquid crystal displays based on a simple physical model”, Opt. 
Eng. 40, 2558 (2001)
[16] I. Moreno, C. Iemmi, A. Márquez, J. Campos, and M. Yzuel, “Modulation light efficiency of diffractive lenses
displayed onto a restricted phase-mostly modulation display”, Appl. Opt. 43, 6278 (2004)
[17] A. Lizana, I. Moreno, A. Márquez, C. Iemmi, E. Fernández, J. Campos, and M. J. Yzuel, “Time fluctuations
of the phase modulation in a liquid crystal on silicon display: characterization and effects in diffractive 
optics,” Opt. Express 16, 16711 (2008)
[18] A. Lizana, A. Márquez, L. Lobato, Y. Rodange, I. Moreno, C. Iemmi, J. Campos ’’The minimum Euclidean
distance principle applied to improve the modulation diffraction efficiency in digitally controlled spatial 
light modulators” Opt. Exp. 18, 10581, (2010)
6


Preparing arbitrary pure states of spatial qudits with a single phase-only spatial light modulator

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23rd Congress of the International Commission for Optics (ICO 23)Journal of Physics: Conference Series 605 (2015) 012035IOP Publishingdoi:10.1088/1742-6596/605/l/012035Preparing arbitrary pure states of spatial qudits with a single phase-only spatial light modulatorJ J M Varga x*, M A Solis-Prosser2, L. Rebón3, A Arias4, LNeves5, C Iemmi1, S Ledesma1 departamento de Física, FCEyN, Universidad de Buenos Aires, Buenos Aires 1428, Argentina 3Center for Optics and Photonics, Universidad de Concepción, Casilla 4016, Concepción, Chile “Instituto de Física de La Plata, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentinaventer for Research, Instituto Tecnológico Metropolitano, A.A 54954, Medellin, Colombia 6Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte, MG 30123-970, BrazilE-mail: miguelQdf .uba. arAbstract. We present a new method for preparing multidimensional spatial qudits by means of a single phase-only spatial light modulator (SLM). This method improves previous ones that use two SLMs, one working in amplitude regime and the other in phase regime. To that end, we addressed diffraction gratings on the slits that define the state and then we performed a spatial filtering in the Fourier plane. The amplitude of the coefficients of the quantum state are determined by the modulation deep of the diffraction gratings, and the relative phase is the mean phase value of the diffraction gratings. This encoding result to be more compact, less expensive and use the photons more efficiently.Quantum systems are the information carriers in quantum information processing and computing protocols. While qubits (two-level systems) are the usual and most basic systems to carry out such tasks, qudits [D-level systems (£>>2)] have been attracting growing interest due to their greater potential for those applications [1]. In particular, single photons are the natural choice for communications since they are easily transportable and have several degrees of freedom (DOFs) to encode information. In the case of using the transverse momentum-position encoding [2, 3], the simplest approach is achieved by discretizing the one-dimensional transverse modes of the photons when they are made to pass through an aperture with D slits which sets the dimension of these so-called spatial qudits.Spatial qudits are relatively simple to generate and offer the possibility of working in high dimensions without cumbersome optical setups. Recently, this encoding has drawn interest for miscellaneous applications such as quantum information protocols [4], quantum games [5], and quantum key distribution [6]. Therefore, the ability to prepare arbitrary pure states of such systems represents an important step toward realizations of quantum optics experiments based on it. Static amplitude and phase masks could be used for this purpose, but in practice it would be extremely difficult and time-consuming as each state intended to be prepared would require to set up the corresponding masks in the apparatus. Programmable spatial light modulators (SLMs) can dramatically simplify this process, allowing real-time manipulations of the state coefficients without physically aligning any optical components. In this regard, it has beenContent from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd 123rd Congress of the International Commission for Optics (ICO 23) IOP Publishingdoi:10.1088/1742-6596/605/l/012035preparationPolarizerSpatial filter/ beam expanderDiode Laser@647 nmND filtersJournal of Physics: Conference Series 605 (2015) 012035(Image plane)(phase)Beam dumpcameraState characterization (Fourier plane) CCD cameraFigure 1. Experimental setupshown [7] that by imaging the output beam of an amplitude-only SLM onto a phase-only SLM allows one to get complete and independent control of the amplitude and phase of the complex coefficients that define the qudit state. However, the use of two SLMs, besides being more expensive, entails two drawbacks: (i) the overall diffraction efficiency is very low, and (ii) in order to avoid even more losses, the image of the first SLM must match the second one pixel by pixel, which is difficult in practice.Here, we present a demonstration of a method for preparing arbitrary pure states of spatial qudits with a single phase-only SLM [8], which has a much higher diffraction efficiency and does not require imaging systems. Among the alternative methods to represent a complex function in a single SLM [9, 10] we choose, for simplicity, a technique where the amplitude information is encoded in a phase-only filter [11, 12], On one hand, the required amplitude is achieved by programming a phase grating, with an appropriate modulation into the slit regions. On the other hand, the required phase value is obtained by adding a constant phase value to the phase grating. Then, the first diffraction order is selected at the Fourier plane and the filtered information is antitransformed on the final plane (Fig. 1). The result is a light distribution corresponding to an image of the slits containing the complete complex modulation. Regarding the type of selected grating, either binary or blazed, the overall quality of the prepared states is very similar. However, blazed gratings provide better diffraction efficiency than binary ones. This enhanced efficiency can be important to increase the signal-to-noise ratio when working with single photon sources.We have analyzed the performance of the proposed method by preparing a large number of arbitrary pure qubit states uniformly distributed on the surface of the Bloch sphere. To quantify the quality of the preparation we used the fidelity between the state intended to be prepared and the density matrix of the state actually prepared which is reconstructed by tomography (Fig. 2). The tomographic process is performed by measurements at different transverse positions in the near and in the far field [13]. In both cases, either binary or blazed grating, we obtained fidelities above 96% and average fidelities of 99.6%. We have found lor // 3 and D = 7 that the fidelitieswere above 94%. All these relevant results are summerized in the Table 1.We compared the luminous efficiency of our setup with the one that employed two SLMs [7]. In the case of using a blazed grating, our scheme is times more efficient, where r/ represents the light attenuation due to polarizing elements needed to obtain pure amplitude modulation223rd Congress of the International Commission for Optics (ICO 23)Journal of Physics: Conference Series 605 (2015) 012035IOP Publishingdoi: 10.1088/1742-6596/605/1/012035Table 1. Average fidelities of preparation for D-level qudits.D 2 3 4 5 7# of states/grating 561 24 70 25 94-/bin 0.996 0.995 0.985 0.968 0.970a>la 0.996 0.996 0.991 0.971 0.977Figure 2. Ten-pixels period gratings: (a) binary and (b) blazed, (c) (Upper panels) amplitude and phase components of the aperture T(x) for preparing the state 0.56|0) +0.83ez0,637r 11). (Lower panels) corresponding phase masks for preparing this state at the first diffraction order, (d) Normalized intensity distributions and theoretical prediction for measurements of this state at near (upper panel) and far field (lower panel), (e) Real and imaginary parts of the experimentally reconstructed density matrix using the blazed grating. The ideal density matrix is shown as a blue wire grid and the obtained fidelity is F = 0.996.with a twisted nematic SLM [15, 16]. The factor tj depends on the parameters involved in each particular case (wavelength, twist angle, birefringence of the SLM, etc.), but a typical value is about 0.1. Therefore, the method presented here appears as a valuable tool for experiments based on this encoding. In addition, it can be useful for assisting the measurement of high-dimensional photonic qudits, also including mixed states, enabling not only the storing of information but also the read out of it. It enables the implementation of the strategy developed in [14] by proyecting the image of the state to be measured at SLM and addresing at this device the phase grating function corresponding to the measurement state. A detection at the center of the filtered diffraction order at the Fourier plane will acomplish the process allowing one to obtain the statistics of arbitrary observables.Finally, it is important to remark that although we have obtained a good quality of323rd Congress of the International Commission for Optics (ICO 23)Journal of Physics: Conference Series 605 (2015) 012035IOP Publishingdoi:10.1088/1742-6596/605/l/012035Table 2. Selected arbitrary statesstate 1 |^i) = 0.70|0> I-0.70c7l-o<>7r|I)state 2 ¡^2) = 0.70|0) + 0.70e*O37r|l)state 3 ¡^3) = —0.69|0> + 073en-357r|l)state 4 = —0.92|0> + 0.40eil’9O’r| 1>state 5 = 0.48|0> + 0.88eiO-S77r11>state 6 \^6) = -0.48|0) + 0.88en'557r|l)time [ms]10.990.9813TS 0.970.960.95tIssrFigure 3. Temporal phase fluctuation model for the SLM.—state 2---- state 3---- state 4---- state 5—state 60 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Phase shift between SLMs [rad/27r]Figure 4. Fidelity vs. phase shift between SLMs, obtained by means of numerical simulation for six arbitrary states (D = 2) using the phase adition method.preparation, the method presents an inherent phase fluctuation associated to the used SLMs, in our case liquid crystal on silicon displays (LCoS). We have selected LCoSs due to their high423rd Congress of the International Commission for Optics (ICO 23) IOP PublishingJournal of Physics: Conference Series 605 (2015) 012035 doi:10.1088/1742-6596/605/l/012035spatial resolution and very high light efficiency. Nevertheless, these displays may lead to flicker in the optical beam because of their addressing scheme (pulse width modulation) which introduces temporal phase fluctuations [17, 18] that affect the quality of the encoded state. In order to study this effect we have analyzed the performance of the proposed method by simulating the preparation of some selected arbitrary states in one SLM and their tomographic reconstruction in a second SLM. For this purpose we have used a simplified model for the typical temporal fluctuations of the SLMs that consists on a triangular shaped fluctuation as is depicted in Fig. 3. Given that the optical configuration contain two similar SLMs that could be desynchronized, we have considered different phase shift between both modulators. Numerical simulations of the fidelity vs. the phase shift are shown in Fig. 4 for six arbitrary states. These states are shown in Table 2 As it can be noted fidelities are almost independent of the phase shift between the phase fluctuations in each SLM. This would indicate that it is not necessary to deal with the temporal synchronization of the modulators.ConclusionsWe have shown that a single phase-only SLM enables the preparation of arbitrary pure states of spatial qudits. Mean values of fidelities of preparation above 94% were obtained and a higher efficiency (at least 10 times) with respect to other methods is expected. It is important to note that this method can be useful not only for preparation but also for assisting the measurement of such systems. It enables the implementation of a strategy based on imaging the state represented on the first SLM onto the second SLM. In this case the first SLM is used to represent the state to be analized and the second one to represent the reconstruction basis. Therefore, the method presented here may become a valuable tool for experiments using spatial qudits.AcknowledgmentWe thank P. Kolenderski and A. Calatayud for helpful discussions. This work was supported by CONICYT PFB08-24 and Milenio ICM P10-030-F (Chile) and by UBACyT 20020100100689, CONICET PIP 112-200801-03047, and ANPCYT PICT 2010-02179 (Argentina).References[1] S. M. Barnett, Quantum Information (Oxford University Press, Oxford, 2009).[2] L. Neves, G. Lima, J. G. Aguirre-Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangledstates of qudits using twin photons”, Phys. Rev. Lett. 94, I0050I (2005).[3] M. N. O’Sullivan-Hale, I. Ali-Khan, R. W. Boyd, and J. C. Howell, “Pixel Entanglement: ExperimentalRealization of Optically Entangled d = 3 and d = 6 Qudits”, Phys. Rev. Lett. 94, 220501 (2005).[4] M. A. Solis-Prosser and L. Neves, “Remote state preparation of spatial qubits”, Phys. Rev. A 84, 012330(2011).[5] P. Kolenderski, U. Sinha, L. Youning, T. Zhao, M. Volpini, A. Cabello, R. Laflamme, and T. Jennewein,“Aharon-Vaidman quantum game with a Young-type photonic qutrit”, Phys. Rev. A 86, 0I232I (2012).[6] S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima, “Quantumkey distribution session with 16-dimensional photonic states”, Sci. Rep. 3, 2316 (2013).[7] 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 (2011).[8] M. A. Solis-Prosser, A. Arias, J. J. M. Varga, L. Rebñ, S. Ledesma, C. Iemmi, and L. 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Preparing arbitrary pure states of spatial qudits with a single phase-only spatial light modulator

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