Abstract

We report minimal quantum state tomography with spatial qubits created by a pair of parametric down converted twin-photons passing through a double-slit. A novel experimental setup is used, which includes a Spatial Light Modulator, as a fundamental tool, to reconstruct the state density matrix. The theory needed to perform a minimal quantum tomography is described. The density matrix is experimentally obtained for the two-qubit photonic states in spatial variables.

© 2010 OSA

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    [Crossref]
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    [Crossref]
  36. L. Neves, G. Lima, E. J. S. Fonseca, L. Davidovich, and S. Pádua, “Characterizing entanglement in qubits created with spatially correlated twin photons,” Phys. Rev. A 76, 032314 (2007).
    [Crossref]
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    [Crossref]
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    [Crossref]
  40. M. R. Barros, R. C. Drumond, W. M. Pimenta, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Optical implementation of minimal Deutsch algorithm.”

2009 (4)

A. Rossi, A. Chiuri, G. Vallone, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[Crossref] [PubMed]

B. C. dos Santos, K. Dechoum, and A. Z. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[Crossref]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. W. Boyd, A. K. Jha, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17, 8287–8293 (2009).
[Crossref] [PubMed]

G. Lima, A. Vargas, L. Neves, R. Guzmán, and C. Saavedra, “Manipulating spatial qudit states with programable optical devices,” Opt. Express 1710688–10696 (2009).
[Crossref] [PubMed]

2008 (2)

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]

A. Rossi, G. Vallone, F. de Martini, and P. Mataloni, “Generation of time-bin entangled photons without temporal post-selection,” Phys. Rev. A 78, 012345 (2008).
[Crossref]

2007 (1)

L. Neves, G. Lima, E. J. S. Fonseca, L. Davidovich, and S. Pádua, “Characterizing entanglement in qubits created with spatially correlated twin photons,” Phys. Rev. A 76, 032314 (2007).
[Crossref]

2006 (4)

E. Yao, S. Franke-Arnold, J. Courtial, M. J. Padgett, and S. M. Barnett, “Observation of quantum entanglement using spatial light modulators,” Opt. Express 14, 13089–13094 (2006).
[Crossref] [PubMed]

A. Ling, K. P. Soh, A. Lamas-Linares, and C. Kurtsiefer, “Experimental polarization state tomography using optimal polarimeters,” Phys. Rev. A 74, 022309 (2006).
[Crossref]

J. P. Home, M. J. McDonnell, D. M. Lucas, G. Imreh, B. C. Keitch, D. J. Szwer, N. R. Thomas, S. C. Webster, D. N. Stacey, and A. M. Steane, “Deterministic entanglement and tomography of ion-spin qubits,” New J. Phys. 8, 188 (2006).
[Crossref]

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schimdt, T. K. Korber, W. Hansel, H. Häffner, C. F. Hoos, and R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref] [PubMed]

2005 (4)

L. Neves, G. Lima, J. G. A. Gómes, 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]

C. Cinelli, M. Barbieri, R. Perris, P. Mataloni, and F. de Martini, “All-versus-nothing nonlocality test of quantum mechanics by two-photon hyperentaglement,” Phys. Rev. Lett. 95240405 (2005).
[Crossref] [PubMed]

T. Yang, Q. Zhang, J. Zhang, J. Yin, Z. Zhao, M. Zukowski, Z.-B. Chen, and J.-W. Pan, “All-versus-nothing violation of local realism by two-photon, four-dimensional entaglement,” Phys. Rev. Lett. 95, 240406 (2005).
[Crossref] [PubMed]

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[Crossref]

2004 (3)

J. Řeháček, B.-G. Englert, and D. Kaszlikowski, “Minimal qubit tomography,” Phys. Rev. A 70, 052321 (2004).
[Crossref]

L. Neves, S. Pádua, and C. Saavedra, “Controlled generation of maximally entangled qudits using twin photons,” Phys. Rev. A 69, 042305 (2004).
[Crossref]

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

2003 (1)

I. Moreno, P. Velásquez, C. R. Fernández-Pousa, and M. M. Sánchez-López, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. 94, 3697–3702 (2003).
[Crossref]

2001 (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

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

2000 (1)

E. J. S. Fonseca, J. C. Machado da Silva, C. H. Monken, and S. Pádua, “Controlling two-particle conditional interference,” Phys. Rev. A 61, 023801 (2000).
[Crossref]

1999 (1)

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally entangled states: production, characterization, and utilization,” Phys. Rev. Lett. 83, 3101–3107 (1999).
[Crossref]

1998 (5)

M. A. Nielsen, E. Knill, and R. Laflamme, “Complete quantum teleportation using nuclear magnetic resonance,” Nature 396, 52–55 (1998).
[Crossref]

R. Derka, V. Bužek, and A. K. Ekert, “Universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement,” Phys. Rev. Lett. 80, 1571–1575 (1998).
[Crossref]

J. I. Latorre, P. Pascual, and R. Tarrach, “Minimal optimal generalized quantum measurements,” Phys. Rev. Lett. 81, 1351–1354 (1998).
[Crossref]

W. K. Wootters, “Entenglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[Crossref]

D. Collins, K. W. Kim, and W. C. Holton, “Deutsch-Jozsa algorithm as a test of quantum computation,” Phys. Rev. A 58, 1633–1636 (1998).
[Crossref]

1997 (1)

P. G. Kwiat, “Hyper-entangled states,” J. Mod. Opt. 44, 2173–2184 (1997).

1995 (1)

S. Massar and S. Popescu, “Optimal extraction of information form finite quantum ensembles,” Phys. Rev. Lett. 74, 1259–1263 (1995).
[Crossref] [PubMed]

1993 (1)

D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Multiparticle interferometry and the superposition principle,” Phys. Today 46, 22–29 (1993).
[Crossref]

1990 (1)

J. G. Rarity and P. R. Tapster, “Experimental violation of Bell’s inequality based on phase and momentum,” Phys. Rev. Lett. 64, 2495–2498 (1990).
[Crossref] [PubMed]

1989 (2)

J. D. Franson, “Bell inequality for position and time,” Phys. Rev. Lett. 62, 2205–2208 (1989).
[Crossref] [PubMed]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1989).
[Crossref]

1988 (2)

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref] [PubMed]

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolky-Rosen-Bohm experiment using pairs of light quanta produced by parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[Crossref] [PubMed]

1855 (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).

Alley, C. O.

Y. H. Shih and C. O. Alley, “New type of Einstein-Podolky-Rosen-Bohm experiment using pairs of light quanta produced by parametric down conversion,” Phys. Rev. Lett. 61, 2921–2924 (1988).
[Crossref] [PubMed]

Barbieri, M.

C. Cinelli, M. Barbieri, R. Perris, P. Mataloni, and F. de Martini, “All-versus-nothing nonlocality test of quantum mechanics by two-photon hyperentaglement,” Phys. Rev. Lett. 95240405 (2005).
[Crossref] [PubMed]

Barnett, S. M.

Barreiro, J. T.

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005).
[Crossref]

Barros, M. R.

W. M. Pimenta, M. R. Barros, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Engineering spatial quantum states of twin-photons.”

M. R. Barros, R. C. Drumond, W. M. Pimenta, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Optical implementation of minimal Deutsch algorithm.”

Bartlett, S. D.

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

Blatt, R.

M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schimdt, T. K. Korber, W. Hansel, H. Häffner, C. F. Hoos, and R. Blatt, “Process tomography of ion trap quantum gates,” Phys. Rev. Lett. 97, 220407 (2006).
[Crossref] [PubMed]

Boyd, R. W.

Brendel, J.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1989).
[Crossref]

Bužek, V.

R. Derka, V. Bužek, and A. K. Ekert, “Universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement,” Phys. Rev. Lett. 80, 1571–1575 (1998).
[Crossref]

Carvalho, M. A. D.

W. M. Pimenta, M. R. Barros, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Engineering spatial quantum states of twin-photons.”

M. R. Barros, R. C. Drumond, W. M. Pimenta, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Optical implementation of minimal Deutsch algorithm.”

Chen, Z.-B.

T. Yang, Q. Zhang, J. Zhang, J. Yin, Z. Zhao, M. Zukowski, Z.-B. Chen, and J.-W. Pan, “All-versus-nothing violation of local realism by two-photon, four-dimensional entaglement,” Phys. Rev. Lett. 95, 240406 (2005).
[Crossref] [PubMed]

Chiuri, A.

A. Rossi, A. Chiuri, G. Vallone, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[Crossref] [PubMed]

Cinelli, C.

C. Cinelli, M. Barbieri, R. Perris, P. Mataloni, and F. de Martini, “All-versus-nothing nonlocality test of quantum mechanics by two-photon hyperentaglement,” Phys. Rev. Lett. 95240405 (2005).
[Crossref] [PubMed]

Collins, D.

D. Collins, K. W. Kim, and W. C. Holton, “Deutsch-Jozsa algorithm as a test of quantum computation,” Phys. Rev. A 58, 1633–1636 (1998).
[Crossref]

Courtial, J.

Cunha, M. T.

M. R. Barros, R. C. Drumond, W. M. Pimenta, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Optical implementation of minimal Deutsch algorithm.”

W. M. Pimenta, M. R. Barros, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Engineering spatial quantum states of twin-photons.”

Dalton, R. B.

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

Davidovich, L.

L. Neves, G. Lima, E. J. S. Fonseca, L. Davidovich, and S. Pádua, “Characterizing entanglement in qubits created with spatially correlated twin photons,” Phys. Rev. A 76, 032314 (2007).
[Crossref]

De Martini, F.

A. Rossi, A. Chiuri, G. Vallone, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[Crossref] [PubMed]

A. Rossi, G. Vallone, F. de Martini, and P. Mataloni, “Generation of time-bin entangled photons without temporal post-selection,” Phys. Rev. A 78, 012345 (2008).
[Crossref]

C. Cinelli, M. Barbieri, R. Perris, P. Mataloni, and F. de Martini, “All-versus-nothing nonlocality test of quantum mechanics by two-photon hyperentaglement,” Phys. Rev. Lett. 95240405 (2005).
[Crossref] [PubMed]

Dechoum, K.

B. C. dos Santos, K. Dechoum, and A. Z. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[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).

Derka, R.

R. Derka, V. Bužek, and A. K. Ekert, “Universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement,” Phys. Rev. Lett. 80, 1571–1575 (1998).
[Crossref]

dos Santos, B. C.

B. C. dos Santos, K. Dechoum, and A. Z. Khoury, “Continuous-variable hyperentanglement in a parametric oscillator with orbital angular momentum,” Phys. Rev. Lett. 103, 230503 (2009).
[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]

Drumond, R. C.

M. R. Barros, R. C. Drumond, W. M. Pimenta, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Optical implementation of minimal Deutsch algorithm.”

Eberhard, P. H.

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally entangled states: production, characterization, and utilization,” Phys. Rev. Lett. 83, 3101–3107 (1999).
[Crossref]

Ekert, A. K.

R. Derka, V. Bužek, and A. K. Ekert, “Universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement,” Phys. Rev. Lett. 80, 1571–1575 (1998).
[Crossref]

Englert, B.-G.

J. Řeháček, B.-G. Englert, and D. Kaszlikowski, “Minimal qubit tomography,” Phys. Rev. A 70, 052321 (2004).
[Crossref]

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M. R. Barros, R. C. Drumond, W. M. Pimenta, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Optical implementation of minimal Deutsch algorithm.”

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[Crossref] [PubMed]

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Other (3)

W. M. Pimenta, M. R. Barros, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Engineering spatial quantum states of twin-photons.”

S. Cialdi, D. Brivio, and M. G. A. Paris, “Demonstration of a programable source of two-photon multiqubit entangled states,” http://www.arxiv.org/abs/quant-ph/0912.2975v3 .

M. R. Barros, R. C. Drumond, W. M. Pimenta, B. Marques, M. A. D. Carvalho, J. Ferraz, M. T. Cunha, and S. Pádua are preparing a manuscript to be called “Optical implementation of minimal Deutsch algorithm.”

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Figures (6)

Fig. 1
Fig. 1

Schematic vector representation of the tomographic strategy. Measurement operators Πj, are proportional to projectors |ψj〉 〈ψj|, with the states |ψj〉 being the tetrahedron vertices of the Bloch sphere.

Fig. 2
Fig. 2

(Color online) Experimental setup scheme for quantum tomography, in the transverse path degrees of freedom of two-qubit photon states. The L1 lens focuses the pump beam in the double slit’s plane; lenses Ls1 and Li1 are used to detect the signal and idler beams at the Fourier plane, while lenses Ls2 and Li2 are used to project the double slits images in the detectors planes. A half-wave plate is placed right after the crystal and polarizers Pi and Ps are positioned in front of APD’s detectors. CNC is a coincidence counter and SLM is the Spatial Light Modulator.

Fig. 3
Fig. 3

Double slit interference patterns. In (a), the conditional interference pattern is presented. Idler detector is kept fixed at xi = 0 (closed circles) or at xi = 250 μm (open circles), while signal detector is scanned. This result is obtained with the SLM turned off. Graphs (b) and (c), show the patterns obtained for phase differences of 0 and π rad, added by the SLM, between states |0〉 and |1〉, respectively, and the relative amplitude ratio necessary for implementing the evolution maps. Plot with closed circles in (a) is reproduced in (b) and (c), as a reference.

Fig. 4
Fig. 4

Double slit images recorded with the idler detector Di fixed at xi = 0, while Ds is scanned in the x direction. Lenses Li2 and Ls2, described in the experimental apparatus, were used. In (a), we have the same experimental parameters used to obtain the conditional interference patterns, shown in Fig. 3(a). The phase difference imposed, by the SLM, on states |0〉 and |1〉 were: (b) 0, (c) π rad The states relative probability attenuation, imposed by the SLM and the polarizers was one half. Closed circles are single counts, and open circles coincidence detections.

Fig. 5
Fig. 5

Double slit conditional images. Closed circles are single counts, and open circles are coincidence detections. In (a) detector Di is fixed in the superior slit, in (b) Di is fixed in the inferior slit, while detector Ds is scanned in the x-direction at the image plane. In these measurements the SLM was turned off.

Fig. 6
Fig. 6

(Color online) Tomographic reconstruction of the output state for two-qubits. The figure represents the modulus of the real and imaginary parts of each density matrix measured element

Equations (5)

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E 1 = [ 1 2 0 0 1 ] , E 2 = [ 1 2 0 0 1 ] , E 3 = [ i 0 0 1 2 ] , E 4 = [ i 0 0 1 2 ] .
c j Tr ( E j ρ E j P ) = Tr ( ρ E j P E j ) ,
c j Tr ( ρ Π j ) .
ρ ^ = ( 0.063 0.131 + i 0.039 0.139 i 0.017 0.010 i 0.003 0.131 i 0.039 0.480 0.388 i 0.051 0.034 + i 0.001 0.139 + i 0.017 0.388 + i 0.051 0.455 0.027 i 0.004 0.010 + i 0.003 0.034 i 0.001 0.027 + i 0.004 0.002 )
F ( Π 1 exp , Π 1 theory ) = 0.991 & F ( Π 2 exp , Π 2 theory ) = 0.995 .

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