Abstract

The state of spatially correlated down-converted photons is usually treated as a two-mode Gaussian entangled state. While intuitively this seems to be reasonable, it is known that new structures in the spatial distributions of these photons can be observed when the phase-matching conditions are properly taken into account. Here, we study how the variances of the near- and far-field conditional probabilities are affected by the phase-matching functions, and we analyze the role of the EPR-criterion regarding the non-Gaussianity and entanglement detection of the spatial two-photon state of spontaneous parametric down-conversion (SPDC). Then we introduce a statistical measure, based on the negentropy of the joint distributions at the near- and far-field planes, which allows for the quantification of the non-Gaussianity of this state. This measure of non-Gaussianity requires only the measurement of the diagonal covariance sub-matrices, and will be relevant for new applications of the spatial correlation of SPDC in CV quantum information processing.

© 2012 Optical Society of America

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  1. J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
    [Crossref] [PubMed]
  2. M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
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  4. E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868 (1999).
    [Crossref]
  5. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. 99, 170408 (2007).
    [Crossref] [PubMed]
  6. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
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  8. A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90, 133603 (2003).
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  9. 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, 100501 (2005).
    [Crossref] [PubMed]
  10. 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).
    [Crossref]
  11. A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  18. S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, “Angular Schmidt modes in spontaneous parametric down-conversion,” Phys. Rev. A 83, 060302 (2011).
    [Crossref]
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    [Crossref]
  20. D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78, 010304 (2008).
    [Crossref]
  21. D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79, 033801 (2009).
    [Crossref]
  22. L. J. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B 42, 114011 (2009).
    [Crossref]
  23. H. Di Lorenzo Pires and M. P. van Exter, “Observation of near-field correlations in spontaneous parametric down-conversion,” Phys. Rev. A 79, 041801 (2009).
    [Crossref]
  24. M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
    [Crossref]
  25. R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, “Quantum entanglement beyond Gaussian criteria,” Proc. Natl. Acad. Sci. U.S.A. 106, 21517 (2009).
    [Crossref] [PubMed]
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    [Crossref]
  27. M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
    [Crossref]
  28. M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A 82, 052341 (2010).
    [Crossref]
  29. M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt. 56, 1829–1837 (2009).
    [Crossref]
  30. H. Di Lorenzo Pires and M. P. van Exter, “Near-field correlations in the two-photon field,” Phys. Rev. A 80, 053820 (2009).
    [Crossref]
  31. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
    [Crossref] [PubMed]
  32. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
    [Crossref]
  33. P. Comon, “Independent component analysis, A new concept?,” Sig. Process. 36, 287–314 (1994).
    [Crossref]
  34. P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high dimensional entangled states,” arXiv:1107.5245v1[quant-ph].
  35. M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
    [Crossref] [PubMed]

2011 (1)

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, “Angular Schmidt modes in spontaneous parametric down-conversion,” Phys. Rev. A 83, 060302 (2011).
[Crossref]

2010 (3)

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A 82, 052341 (2010).
[Crossref]

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

2009 (8)

H. Di Lorenzo Pires, C. H. Monken, and M. P. van Exter, “Direct measurement of transverse-mode entanglement in two-photon states,” Phys. Rev. A 80, 022307 (2009).
[Crossref]

M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt. 56, 1829–1837 (2009).
[Crossref]

H. Di Lorenzo Pires and M. P. van Exter, “Near-field correlations in the two-photon field,” Phys. Rev. A 80, 053820 (2009).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79, 033801 (2009).
[Crossref]

L. J. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B 42, 114011 (2009).
[Crossref]

H. Di Lorenzo Pires and M. P. van Exter, “Observation of near-field correlations in spontaneous parametric down-conversion,” Phys. Rev. A 79, 041801 (2009).
[Crossref]

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, “Quantum entanglement beyond Gaussian criteria,” Proc. Natl. Acad. Sci. U.S.A. 106, 21517 (2009).
[Crossref] [PubMed]

2008 (2)

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78, 010304 (2008).
[Crossref]

2007 (3)

S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007).
[Crossref]

K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A 75, 050101 (2007).
[Crossref]

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. 99, 170408 (2007).
[Crossref] [PubMed]

2006 (1)

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[Crossref] [PubMed]

2005 (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, 100501 (2005).
[Crossref] [PubMed]

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).
[Crossref]

2004 (3)

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[Crossref]

C. K. Law and J. H. Eberly, “Analysis and Interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[Crossref] [PubMed]

2003 (1)

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90, 133603 (2003).
[Crossref] [PubMed]

2002 (1)

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

1999 (1)

E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868 (1999).
[Crossref]

1998 (1)

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[Crossref]

1995 (1)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref] [PubMed]

1994 (2)

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 505122 (1994).
[Crossref] [PubMed]

P. Comon, “Independent component analysis, A new concept?,” Sig. Process. 36, 287–314 (1994).
[Crossref]

1985 (1)

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409 (1985).
[Crossref] [PubMed]

1935 (1)

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

Abouraddy, A. F.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. 99, 170408 (2007).
[Crossref] [PubMed]

Aguirre Gómez, J. G.

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, 100501 (2005).
[Crossref] [PubMed]

Ali Khan, I.

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).
[Crossref]

Andersen, U. L.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Bachor, H. A.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Banaszek, K.

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[Crossref]

Barnett, S. M.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

Bennink, R. S.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

Bentley, S. J.

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

Bobrov, I. B.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, “Angular Schmidt modes in spontaneous parametric down-conversion,” Phys. Rev. A 83, 060302 (2011).
[Crossref]

Bowen, W. P.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Boyd, R.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

Boyd, R. W.

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).
[Crossref]

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

Brambilla, E.

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90, 133603 (2003).
[Crossref] [PubMed]

Cavalcanti, E. G.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Chan, K. W.

K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A 75, 050101 (2007).
[Crossref]

Cirac, J. I.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[Crossref] [PubMed]

Comon, P.

P. Comon, “Independent component analysis, A new concept?,” Sig. Process. 36, 287–314 (1994).
[Crossref]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[Crossref]

D’Angelo, M.

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[Crossref]

de Matos Filho, R. L.

S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007).
[Crossref]

Di Lorenzo Pires, H.

H. Di Lorenzo Pires, C. H. Monken, and M. P. van Exter, “Direct measurement of transverse-mode entanglement in two-photon states,” Phys. Rev. A 80, 022307 (2009).
[Crossref]

H. Di Lorenzo Pires and M. P. van Exter, “Near-field correlations in the two-photon field,” Phys. Rev. A 80, 053820 (2009).
[Crossref]

H. Di Lorenzo Pires and M. P. van Exter, “Observation of near-field correlations in spontaneous parametric down-conversion,” Phys. Rev. A 79, 041801 (2009).
[Crossref]

Dixon, P. B.

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high dimensional entangled states,” arXiv:1107.5245v1[quant-ph].

Drummond, P. D.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Eberly, J. H.

K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A 75, 050101 (2007).
[Crossref]

C. K. Law and J. H. Eberly, “Analysis and Interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[Crossref] [PubMed]

Einstein, A.

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

Eisert, J.

M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt. 56, 1829–1837 (2009).
[Crossref]

Ether, D. S.

S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007).
[Crossref]

Fonseca, E. J. S.

E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868 (1999).
[Crossref]

Franke-Arnold, S.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

Gatti, A.

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90, 133603 (2003).
[Crossref] [PubMed]

Genoni, M. G.

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A 82, 052341 (2010).
[Crossref]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[Crossref]

Giedke, G.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[Crossref] [PubMed]

Giovannetti, V.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

Gomes, R. M.

R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, “Quantum entanglement beyond Gaussian criteria,” Proc. Natl. Acad. Sci. U.S.A. 106, 21517 (2009).
[Crossref] [PubMed]

Henkel, C.

M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt. 56, 1829–1837 (2009).
[Crossref]

Hong, C. K.

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409 (1985).
[Crossref] [PubMed]

Howell, J. C.

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).
[Crossref]

J. C. Howell, R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion,” Phys. Rev. Lett. 92, 210403 (2004).
[Crossref] [PubMed]

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high dimensional entangled states,” arXiv:1107.5245v1[quant-ph].

Howland, G. A.

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high dimensional entangled states,” arXiv:1107.5245v1[quant-ph].

Hyvärinen, A.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[Crossref]

Ivanov, D. P.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, “Angular Schmidt modes in spontaneous parametric down-conversion,” Phys. Rev. A 83, 060302 (2011).
[Crossref]

Jack, B.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

Jha, A. K.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

Kalinkin, A. A.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, “Angular Schmidt modes in spontaneous parametric down-conversion,” Phys. Rev. A 83, 060302 (2011).
[Crossref]

Karhunen, J.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[Crossref]

Kim, Y. H.

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[Crossref]

Klyshko, D. N.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 505122 (1994).
[Crossref] [PubMed]

Korn, D.

M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt. 56, 1829–1837 (2009).
[Crossref]

Kulik, S. P.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, “Angular Schmidt modes in spontaneous parametric down-conversion,” Phys. Rev. A 83, 060302 (2011).
[Crossref]

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[Crossref]

Lam, P. K.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Law, C. K.

C. K. Law and J. H. Eberly, “Analysis and Interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[Crossref] [PubMed]

Leach, J.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

Leuchs, G.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Lima, G.

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, 100501 (2005).
[Crossref] [PubMed]

Lugiato, L. A.

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90, 133603 (2003).
[Crossref] [PubMed]

Lundeen, J. S.

L. J. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B 42, 114011 (2009).
[Crossref]

Mancini, S.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

Mandel, L.

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409 (1985).
[Crossref] [PubMed]

Monken, C. H.

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

H. Di Lorenzo Pires, C. H. Monken, and M. P. van Exter, “Direct measurement of transverse-mode entanglement in two-photon states,” Phys. Rev. A 80, 022307 (2009).
[Crossref]

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, 100501 (2005).
[Crossref] [PubMed]

E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868 (1999).
[Crossref]

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[Crossref]

Neves, L.

L. J. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B 42, 114011 (2009).
[Crossref]

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, 100501 (2005).
[Crossref] [PubMed]

O’Sullivan-Hale, M. N.

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).
[Crossref]

Oja, E.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[Crossref]

Ostermeyer, M.

M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt. 56, 1829–1837 (2009).
[Crossref]

Padgett, M. J.

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

Pádua, S.

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

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, 100501 (2005).
[Crossref] [PubMed]

E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868 (1999).
[Crossref]

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[Crossref]

Paris, M. G. A.

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A 82, 052341 (2010).
[Crossref]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[Crossref]

Pellat-Finet, P.

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79, 033801 (2009).
[Crossref]

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref] [PubMed]

Podolsky, B.

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

Puhlmann, D.

M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt. 56, 1829–1837 (2009).
[Crossref]

Reid, M. D.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Ribeiro, P. H. S.

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79, 033801 (2009).
[Crossref]

R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, “Quantum entanglement beyond Gaussian criteria,” Proc. Natl. Acad. Sci. U.S.A. 106, 21517 (2009).
[Crossref] [PubMed]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78, 010304 (2008).
[Crossref]

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[Crossref]

Rosen, N.

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

Rubin, M. H.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 505122 (1994).
[Crossref] [PubMed]

Saavedra, C.

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, 100501 (2005).
[Crossref] [PubMed]

Saleh, B. E. A.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. 99, 170408 (2007).
[Crossref] [PubMed]

Salles, A.

R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, “Quantum entanglement beyond Gaussian criteria,” Proc. Natl. Acad. Sci. U.S.A. 106, 21517 (2009).
[Crossref] [PubMed]

Schneeloch, J.

P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high dimensional entangled states,” arXiv:1107.5245v1[quant-ph].

Sergienko, A. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref] [PubMed]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 505122 (1994).
[Crossref] [PubMed]

Shih, Y.

M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004).
[Crossref]

Shih, Y. H.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref] [PubMed]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 505122 (1994).
[Crossref] [PubMed]

Straupe, S. S.

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, “Angular Schmidt modes in spontaneous parametric down-conversion,” Phys. Rev. A 83, 060302 (2011).
[Crossref]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref] [PubMed]

Tasca, D. S.

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79, 033801 (2009).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78, 010304 (2008).
[Crossref]

Teich, M. C.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. 99, 170408 (2007).
[Crossref] [PubMed]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[Crossref]

Tombesi, P.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

Torres, J. P.

K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A 75, 050101 (2007).
[Crossref]

Toscano, F.

R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, “Quantum entanglement beyond Gaussian criteria,” Proc. Natl. Acad. Sci. U.S.A. 106, 21517 (2009).
[Crossref] [PubMed]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79, 033801 (2009).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78, 010304 (2008).
[Crossref]

van Exter, M. P.

H. Di Lorenzo Pires, C. H. Monken, and M. P. van Exter, “Direct measurement of transverse-mode entanglement in two-photon states,” Phys. Rev. A 80, 022307 (2009).
[Crossref]

H. Di Lorenzo Pires and M. P. van Exter, “Observation of near-field correlations in spontaneous parametric down-conversion,” Phys. Rev. A 79, 041801 (2009).
[Crossref]

H. Di Lorenzo Pires and M. P. van Exter, “Near-field correlations in the two-photon field,” Phys. Rev. A 80, 053820 (2009).
[Crossref]

Vitali, D.

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

Walborn, S. P.

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79, 033801 (2009).
[Crossref]

R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, “Quantum entanglement beyond Gaussian criteria,” Proc. Natl. Acad. Sci. U.S.A. 106, 21517 (2009).
[Crossref] [PubMed]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78, 010304 (2008).
[Crossref]

S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007).
[Crossref]

Walmsley, I. A.

L. J. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B 42, 114011 (2009).
[Crossref]

Wolf, M. M.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[Crossref] [PubMed]

Yarnall, T.

T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. 99, 170408 (2007).
[Crossref] [PubMed]

Zagury, N.

S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007).
[Crossref]

Zhang, L. J.

L. J. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B 42, 114011 (2009).
[Crossref]

J. Mod. Opt. (1)

M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt. 56, 1829–1837 (2009).
[Crossref]

J. Phys. B (1)

L. J. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B 42, 114011 (2009).
[Crossref]

Phys. Rep. (1)

S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010).
[Crossref]

Phys. Rev. (1)

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

Phys. Rev. A (14)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref] [PubMed]

C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409 (1985).
[Crossref] [PubMed]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 505122 (1994).
[Crossref] [PubMed]

C. H. Monken, P. H. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123 (1998).
[Crossref]

H. Di Lorenzo Pires, C. H. Monken, and M. P. van Exter, “Direct measurement of transverse-mode entanglement in two-photon states,” Phys. Rev. A 80, 022307 (2009).
[Crossref]

K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A 75, 050101 (2007).
[Crossref]

S. S. Straupe, D. P. Ivanov, A. A. Kalinkin, I. B. Bobrov, and S. P. Kulik, “Angular Schmidt modes in spontaneous parametric down-conversion,” Phys. Rev. A 83, 060302 (2011).
[Crossref]

S. P. Walborn, D. S. Ether, R. L. de Matos Filho, and N. Zagury, “Quantum teleportation of the angular spectrum of a single-photon field,” Phys. Rev. A 76, 033801 (2007).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78, 010304 (2008).
[Crossref]

D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79, 033801 (2009).
[Crossref]

H. Di Lorenzo Pires and M. P. van Exter, “Observation of near-field correlations in spontaneous parametric down-conversion,” Phys. Rev. A 79, 041801 (2009).
[Crossref]

H. Di Lorenzo Pires and M. P. van Exter, “Near-field correlations in the two-photon field,” Phys. Rev. A 80, 053820 (2009).
[Crossref]

M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A 78, 060303 (2008).
[Crossref]

M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A 82, 052341 (2010).
[Crossref]

Phys. Rev. Lett. (11)

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006).
[Crossref] [PubMed]

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002).
[Crossref] [PubMed]

C. K. Law and J. H. Eberly, “Analysis and Interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. 92, 127903 (2004).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90, 133603 (2003).
[Crossref] [PubMed]

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, 100501 (2005).
[Crossref] [PubMed]

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).
[Crossref]

A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010).
[Crossref] [PubMed]

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[Crossref]

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Figures (5)

Fig. 1
Fig. 1 The far- and near-field conditional probability distributions, while considering the state of SPDC ( p F F S and p N F S) and when some Gaussian approximations are assumed ( p F F , α i G and p N F , α i G). In (a)–(d) the curves p F F S and p N F S (red solid lines) are compared with p F F , α 2 G and p N F , α 2 G for α1 = 0.45 (green dashed line), α2 = 0.72 (blue dot-dashed line), and for some specific values of P. In (e) and (f) the variances of the momentum and position conditional probabilities are plotted in terms of P, respectively. α3 = 1.

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Fig. 2
Fig. 2 The EPR-criterion plotted in terms of P. The red (solid) curve corresponds to the EPR values of the two-photon state generated in the SPDC [Eq. (1)]. The green (dotted), blue (dot-dashed) and black (dotted) curves describe the EPR-criterion for the two-photon Gaussian states defined in terms of α1, α2 and α3, respectively.

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Fig. 3
Fig. 3 In (a)[(b)] the negentropies of the near- and far-field conditional (marginal) distributions are plotted in terms of P. In (c) [(d)] we show the non-Gaussianity of the conditional (marginal) distributions.

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Fig. 4
Fig. 4 Comparison of the Mancini-criterion while considering the state of SPDC (red solid line) and when some Gaussian approximations are adopted. Here we use α1 = 0.45 (green dashed line), α2 = 0.72 (blue dot-dashed line), and α3 = 1 (black dotted line) to describe the spatial Gaussian approximations [See Eq. (5) and Eq. (6) of main paper].

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Fig. 5
Fig. 5 Here we show the non-Gaussianity of the marginal distributions for larger values of P.

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Equations (98)

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| Ψ 〉 ∝ ∬ d q 1 d q 2 e − ζ 4 | q 1 + q 2 | 2 sinc L | q 1 − q 2 | 2 4 k p | q 1 , q 2 〉 ,
| Ψ 〉 ∝ ∬ d ρ 1 d ρ 2 e − 1 4 ζ | ρ 1 + ρ 2 | 2 sint k p | ρ 1 − ρ 2 | 2 4 L | ρ 1 , ρ 2 〉 ,
p F F S ( q ˜ 1 , q ˜ 2 )           ∝             e − 1 2 ( q ˜ 1 + q ˜ 2 ) 2 ( sinc P 2 ( q ˜ 1 − q ˜ 2 ) 2 4 ) 2 ,
p N F S ( x ˜ 1 , x ˜ 2 )           ∝             e − 1 2 σ 2 ( x ˜ 1 + x ˜ 2 ) 2 ( sint ( x ˜ 1 − x ˜ 2 ) 2 4 P 2 ) 2 ,
p F F , α i G ( q ˜ 1 , q ˜ 2 )           ∝           e − 1 2 ( q ˜ 1 + q ˜ 2 ) 2 e − α i P 2 2 ( q ˜ 1 − q ˜ 2 ) 2 ,
p N F , α i G ( x ˜ 1 , x ˜ 2 )             ∝             e − 1 2 σ 2 ( x ˜ 1 + x ˜ 2 ) 2 e − 1 2 α i P 2 ( x ˜ 1 − x ˜ 2 ) 2 ,
n G T ≡ N [ p F F S ( q ˜ 1 , q ˜ 2 ) ] + N [ p N F S ( x ˜ 1 , x ˜ 2 ) ] ,
n G T ≈ n G C + n G M       ( iff   P ≪ 1 ) .
| Ψ 〉 = ∫ d q 1 ∫ d q 2 Ψ ( q 1 , q 2 ) | 1 q 1 〉 | 1 q 2 〉 ,
Ψ ( q 1 , q 2 ) ∝ Δ ( q 1 + q 2 ) Θ ( q 1 − q 2 ) ,
Ψ ( q 1 , q 2 ) ∝ e − ( q 1 + q 2 ) 2 / 2 σ + 2 e − ( q 1 − q 2 ) 2 / 2 δ − 2 ,
Ψ ( x 1 , x 2 ) ∝ e − σ + 2 ( x 1 + x 2 ) 2 / 8 e − δ − 2 ( x 1 − x 2 ) 2 / 8 .
P ( x 1 | x 2 = 0 ) = σ + 2 + δ − 2 2 π   exp   [ − σ + 2 + δ − 2 4 x 1 2 ] ,
P ( q 1 | q 2 = 0 ) = σ + 2 + δ − 2 π σ + 2 δ − 2   exp   [ − σ + 2 + δ − 2 σ + 2 δ − 2 q 1 2 ] .
( Δ x 1 | x 2 ) 2 = 2 σ + 2 + δ − 2 ,
( Δ q 1 | q 2 ) 2 = σ + 2 δ − 2 2 ( σ + 2 + δ − 2 ) .
( Δ x 1 | x 2 ) 2 ( Δ q 1 | q 2 ) 2 = σ + 2 δ − 2 ( σ + 2 + δ − 2 ) 2 .
K = 1 4 ( σ + δ − + δ − σ + ) 2 = 1 4 ( σ + 2 + δ − 2 ) 2 σ + 2 δ − 2 .
( Δ x 1 | x 2 ) 2 ( Δ q 1 | q 2 ) 2 = 1 4 K .
[ Δ ( x ˜ 1 − x ˜ 2 ) ] 2 [ Δ ( q ˜ 1 + q ˜ 2 ) ] 2 ≥ 1.
[ Δ ( q ˜ 1 + q ˜ 2 ) ] 2 ∝ ∫ d q ˜ + q ˜ + 2 e − 1 2 q ˜ + 2 = 1 ,
[ Δ ( x ˜ 1 − x ˜ 2 ) ] 2 ∝ ∫ d x ˜ − x ˜ − 2 sint 2 1 4 P x ˜ − 2 = 4 A 2 A 1 P 2 ,
A 1 = ∫ − ∞ ∞ d ξ   sint 2 ξ 2 ≈ 1.4008 ,
A 2 = ∫ − ∞ ∞ d ξ   ξ 2   sint 2 ξ 2 ≈ 0.5897.
N ≡ H [ p G ˜ ( ξ 1 , ξ 2 ) ] − H [ p ( ξ 1 , ξ 2 ) ] ,
H [ p ( ξ 1 , ξ 2 ) ] ≡ − ∫ d ξ 1 ∫ d ξ 2 p ( ξ 1 , ξ 2 ) log 2 ( p ( ξ 1 , ξ 2 ) ) .
p F F S ( q ˜ 1 , q ˜ 2 ) = 3 P 4 2 π e − ( q ˜ 1 + q ˜ 2 ) 2 2 sinc 2 P 2 4 ( q ˜ 1 − q ˜ 2 ) 2 .
Λ F F S = ( ( Δ q ˜ 1 ) S 2 〈 q ˜ 1 q ˜ 2 〉 S − 〈 q ˜ 1 〉 S 〈 q ˜ 2 〉 S 〈 q ˜ 2 q ˜ 1 〉 S − 〈 q ˜ 2 〉 S 〈 q ˜ 1 〉 S 〈 Δ q ˜ 2 〉 S 2 ) .
( Δ q ˜ 1 ) S 2 = ( Δ q ˜ 2 ) S 2 = ∬ d q ˜ 1 d q ˜ 2 q ˜ 1 2 p F F S ( q ˜ 1 , q ˜ 2 ) = 1 4 ( 1 + 3 P 2 ) ,
〈 q ˜ 1 q ˜ 2 〉 S = 〈 q ˜ 2 q ˜ 1 〉 S = ∬ d q ˜ 1 d q ˜ 2 q ˜ 1 q ˜ 2 p F F S ( q ˜ 1 , q ˜ 2 ) = 1 4 ( 1 − 3 P 2 ) .
p F F G ( q ˜ 1 , q ˜ 2 ) = 1 π δ − e − ( q ˜ 1 + q ˜ 2 ) 2 2 e − ( q ˜ 1 − q ˜ 2 ) 2 2 δ − 2 .
( Δ q ˜ 1 ) G 2 = ( Δ q ˜ 2 ) G 2 = ∬ d q ˜ 1 d q ˜ 2 q ˜ 1 2 p F F G ( q ˜ 1 , q ˜ 2 ) = 1 4 ( 1 + δ − 2 ) ,
〈 q ˜ 1 q ˜ 2 〉 G = 〈 q ˜ 2 q ˜ 1 〉 G = ∬ d q ˜ 1 d q ˜ 2 q ˜ 1 q ˜ 2 p F F G ( q ˜ 1 , q ˜ 2 ) = 1 4 ( 1 − δ − 2 ) .
δ − 2 = 3 P 2 ,
H [ p F F G ˜ ( q ˜ 1 , q ˜ 2 ) ] = log 2 π e 3 P .
H [ p F F S ( q ˜ 1 , q ˜ 2 ) ] = − ∬ d q ˜ 1 d q ˜ 2 p F F S ( q ˜ 1 , q ˜ 2 ) log 2 ( p F F S ( q ˜ 1 , q ˜ 2 ) ) = log 2 4 2 π 3 P − 3 P 4 2 π ∬ d q ˜ 1 d q ˜ 2 { e − ( q ˜ 1 + q ˜ 2 ) 2 2 × [ log 2 e − ( q ˜ 1 + q ˜ 2 ) 2 2 + log 2 sinc 2 P 2 4 ( q ˜ 1 − q ˜ 2 ) 2 ] } .
H [ p F F S ( q ˜ 1 , q ˜ 2 ) ] = log 2 4 2 π 3 P 1 2 ln 2 × [ 1 − 3 π ∫ 0 ∞ d v   sinc 2 v 2 ln ( sinc 2 v 2 ) ] .
∫ 0 ∞ d v   sinc 2 v 2 ln ( sinc 2 v 2 ) ≈ − 0.364.
H [ p F F S ( q ˜ 1 , q ˜ 2 ) ] ≈ log 2 4 2 π 3 P + 1.17.
N [ p S ( q ˜ 1 , q ˜ 2 ) ] = H [ p F F G ˜ ( q ˜ 1 , q ˜ 2 ) ] − H [ p F F S ( q ˜ 1 , q ˜ 2 ) ] ≈ log 2 π e 3 P − log 2 4 2 π 3 P − 1.17 ≈ log 2 e 27 32 − 1.17 ≈ 0.15.
p N F S ( x ˜ 1 , x ˜ 2 ) = 1 C e − ( x ˜ 1 + x ˜ 2 ) 2 2 σ sint 2 ( x ˜ 1 − x ˜ 2 ) 2 4 P 2 ,
C = 2 π A 1 σ P .
Λ N F S = ( ( Δ x ˜ 1 ) S 2 〈 x ˜ 1 x ˜ 2 〉 S 〈 x ˜ 2 x ˜ 1 〉 S ( Δ x ˜ 2 ) S 2 ) ,
( Δ x ˜ 1 ) S 2 = ( Δ x ˜ 2 ) S 2 = ∬ d x ˜ 1 d x ˜ 2 x ˜ 1 2 p N F S ( x ˜ 1 , x ˜ 2 ) = 1 4 ( σ + 4 A 2 P 2 A 1 ) ,
〈 x ˜ 1 x ˜ 2 〉 S = 〈 x ˜ 2 x ˜ 1 〉 S = ∬ d x ˜ 1 d x ˜ 2 x ˜ 1 x ˜ 2 p N F S ( x ˜ 1 , x ˜ 2 ) = 1 4 ( σ − 4 A 2 P 2 A 1 ) .
p N F G ( x ˜ 1 , x ˜ 2 ) = 1 π σ τ e − ( x ˜ 1 + x ˜ 2 ) 2 2 σ e − ( x ˜ 1 − x ˜ 2 ) 2 2 τ 2 .
Λ N F G = ( ( Δ x ˜ 1 ) G 2 〈 x ˜ 1 x ˜ 2 〉 G 〈 x ˜ 2 x ˜ 1 〉 G ( Δ x ˜ 2 ) G 2 ) ,
( Δ x ˜ 1 ) G 2 = ( Δ x ˜ 2 ) G 2 = ∬ d x ˜ 1 d x ˜ 2 x ˜ 1 2 p N F G ( x ˜ 1 , x ˜ 2 ) = 1 4 ( σ + τ 2 ) ,
〈 x ˜ 1 x ˜ 2 〉 G = 〈 x ˜ 2 x ˜ 1 〉 G = ∬ d x ˜ 1 d x ˜ 2 x ˜ 1 x ˜ 2 p N F G ( x ˜ 1 , x ˜ 2 ) = 1 4 ( σ − τ 2 ) .
τ 2 = 4 A 2 P 2 A 1 ,
H [ p N F G ˜ ( x ˜ 1 , x ˜ 2 ) ] = log 2 ( 2 π e σ A 2 A 1 P ) .
H [ p N F S ( x ˜ 1 , x ˜ 2 ) ] = − ∬ d x ˜ 1 d x ˜ 2 p N F S ( x ˜ 1 , x ˜ 2 ) log 2 ( p N F S ( x ˜ 1 , x ˜ 2 ) ) = log 2 ( 2 π σ A 1 P ) − 1 2 π σ A 1 P ∬ d x ˜ 1 d x ˜ 2 { e − ( x ˜ 1 + x ˜ 2 ) 2 2 σ × sint 2 ( x ˜ 1 − x ˜ 2 ) 2 4 P 2 [ log 2 e − ( x ˜ 1 + x ˜ 2 ) 2 2 σ + log 2 sint 2 ( x ˜ 1 − x ˜ 2 ) 2 4 P 2 ] } .
H [ p N F S ( x ˜ 1 , x ˜ 2 ) ] = log 2 ( 2 π σ A 1 P ) + 1 2 ln 2 [ 1 − 2 A 1 ∫ d s     sint 2 s 2 ln   sint 2 s 2 ] .
I = ∫ d s   sint 2 s 2 ln   sint 2 s 2 ,
H [ p N F S ( x ˜ 1 , x ˜ 2 ) ] ≈ log 2 ( 2 π σ A 1 P ) + 1.434.
N [ p N F S ( x ˜ 1 , x ˜ 2 ) ] = H [ p N F G ˜ ( x ˜ 1 , x ˜ 2 ) ] − H [ p N F S ( x ˜ 1 , x ˜ 2 ) ] ≈ log 2 ( 2 π e σ A 2 A 1 P ) − log 2 ( 2 π σ A 1 P ) − 1.434 ≈ log 2 ( 2 π e A 2 A 1 3 ) − 1.434 ≈ 0.22.
n G T ≈ 0.15 + 0.22 = 0.37.
n G T = N N F + N F F = 0 ,
∫ d q ˜ 1 d q ˜ 2 p F F G ˜ ( q ˜ 1 , q ˜ 2 ) log 2 p F F G ˜ ( q ˜ 1 , q ˜ 2 ) = ∫ d q ˜ 1 d q ˜ 2 p F F S ( q ˜ 1 , q ˜ 2 ) log 2 p F F S ( q ˜ 1 , q ˜ 2 ) ,
∫ d x ˜ 1 d x ˜ 2 p N F G ˜ ( x ˜ 1 , x ˜ 2 ) log 2 p N F G ˜ ( x ˜ 1 , x ˜ 2 ) = ∫ d x ˜ 1 d x ˜ 2 p N F S ( x ˜ 1 , x ˜ 2 ) log 2 p N F S ( x ˜ 1 , x ˜ 2 ) .
p F F G ˜ ( q ˜ 1 , q ˜ 2 ) log 2 p F F G ˜ ( q ˜ 1 , q ˜ 2 ) = p F F S ( q ˜ 1 , q ˜ 2 ) log 2 p F F S ( q ˜ 1 , q ˜ 2 ) ,
p N F G ˜ ( x ˜ 1 , x ˜ 2 ) log 2 p N F G ˜ ( x ˜ 1 , x ˜ 2 ) = p N F S ( x ˜ 1 , x ˜ 2 ) log 2 p N F S ( x ˜ 1 , x ˜ 2 ) .
p F F G ˜ ( q ˜ 1 , q ˜ 2 ) = p F F S ( q ˜ 1 , q ˜ 2 ) ,
p N F G ˜ ( x ˜ 1 , x ˜ 2 ) = p N F S ( x ˜ 1 , x ˜ 2 ) .
N [ p N F S ( x ˜ 1 , x ˜ 2 ) ] = N [ p F F S ( q ˜ 1 , q ˜ 2 ) ] = 0.
n G T = 0.
Λ 11 = 〈 ξ 1 2 〉 − 〈 ξ 1 〉 2 = ∫ d ξ 1   ξ 1 2 p ( ξ 1 ) − ( ∫ d ξ 1   ξ 1 p ( ξ 1 ) ) 2 ,
Λ 22 = 〈 ξ 2 2 〉 − 〈 ξ 2 〉 2 = ∫ d ξ 2   ξ 2 2 p ( ξ 2 ) − ( ∫ d ξ 2   ξ 2 p ( ξ 2 ) ) 2 ,
Λ 12 = 〈 ξ 1 ξ 2 〉 − 〈 ξ 1 〉 〈 ξ 2 〉 = ∫ d ξ 1   ξ 1 p ( ξ 1 ) ∫ d ξ 2   ξ 2 p ( ξ 2 ) − ∫ d ξ 1   ξ 1 p ( ξ 1 ) ∫ d ξ 2   ξ 2 p ( ξ 2 ) = 0 ,
Λ 21 = Λ 12 = 0 ,
p G ˜ ( ξ 1 , ξ 2 ) = 1 2 π det Λ exp [ − 1 2 ( ξ − μ ) Λ − 1 ( ξ − μ ) T ] = e − ( ξ 1 − 〈 ξ 1 〉 ) 2 2 Λ 11 2 π Λ 11 × e − ( ξ 2 − 〈 ξ 2 〉 ) 2 2 Λ 22 2 π Λ 22 = p G ˜ ( ξ 1 ) p G ˜ ( ξ 2 ) ,
H [ p ( ξ 1 , ξ 2 ) ] = H [ p ( ξ 1 ) ] + H [ p ( ξ 2 ) ] ,
N [ p ( ξ 1 , ξ 2 ) ] = H [ p G ˜ ( ξ 1 , ξ 2 ) ] − H [ p ( ξ 1 , ξ 2 ) ] = H [ p G ˜ ( ξ 1 ) ] − H [ p ( ξ 1 ) ] + H [ p G ˜ ( ξ 2 ) ] − H [ p ( ξ 2 ) ] = N [ p ( ξ 1 ) ] + N [ p ( ξ 2 ) ] .
n G T = N [ p F F S ( q ˜ 1 , q ˜ 2 ) ] + N [ p N F S ( x ˜ 1 , x ˜ 2 ) ] = N [ p F F S ( q ˜ 1 ) ] + N [ p F F S ( q ˜ 2 ) ] + N [ p N F S ( x ˜ 1 ) ] + N [ p N F S ( x ˜ 2 ) ] = n G M 1 + n G M 2 ,
n G T = 2 × n G M ,
p F F S ( q ˜ 2 ) = ∫ d q ˜ 1 p F F S ( q ˜ 1 , q ˜ 2 ) ,
p N F S ( x ˜ 2 ) = ∫ d x ˜ 1 p N F S ( x ˜ 1 , x ˜ 2 ) .
p F F S ( q ˜ 2 ) ∝ ∫ d u e − 2 P 2 ( u + P q ˜ 2 ) 2 sinc 2 u 2 ,
p N F S ( x ˜ 2 ) ∝ ∫ d v e − 2 P 2 ( v + x ˜ 2 P ) 2 sint 2 v 2 .
p F F S ( q ˜ 2 ) ∝ ∫ d u δ ( u + P q ˜ 2 ) sinc 2 u 2 ∝ sinc 2 ( P 2 q ˜ 2 ) ,
p N F S ( x ˜ 2 ) ∝ e − 2 x ˜ 2 2 .
N [ p F F S ( q ˜ 2 ) ] ≈ log 2 ( 27 e 32 ) − 0.4447 = 0.154.
p F F S ( q ˜ 2 ) ∝ e − 2 q ˜ 2 2 ,
p N F S ( x ˜ 2 ) ∝ ∫ d v δ ( v + x ˜ 2 P ) sint 2 v 2 ∝ sint 2 x ˜ 2 2 P 2 .
N [ p N F S ( x ˜ 2 ) ] ≈ log 2 ( 2 π e A 2 A 1 3 ) − 0.7122 = 0.224
V = ( 1 4 ( 1 + 4 A 2 P 2 A 1 ) 0 1 4 ( 1 − 4 A 2 P 2 A 1 ) 0 0 1 4 ( 1 + 3 P 2 ) 0 1 4 ( 1 − 3 P 2 ) 1 4 ( 1 − 4 A 2 P 2 A 1 ) 0 1 4 ( 1 + 4 A 2 P 2 A 1 ) 0 0 1 4 ( 1 − 3 P 2 ) 0 1 4 ( 1 + 3 P 2 ) ) .
N [ p F F S ( q ˜ 1 , q ˜ 2 ) ] = H [ p F F G ˜ ( q ˜ 1 , q ˜ 2 ) ] − H [ p F F S ( q ˜ 1 , q ˜ 2 ) ] .
N [ p F F S ( q ˜ 1 , q ˜ 2 ) ] = H [ p F F G ˜ ( q ˜ 1 , q ˜ 2 ) ] + H [ p F F G ˜ ( q ˜ 2 ) ] − H [ p F F S ( q ˜ 1 | q ˜ 2 ) ] − H [ p F F S ( q ˜ 2 ) ] = H [ p F F G ˜ ( q ˜ 1 | q ˜ 2 ) ] − H [ p F F S ( q ˜ 1 | q ˜ 2 ) ] + N [ p F F S ( q ˜ 2 ) ] .
p F F S ( q ˜ 1 | q ˜ 2 = 0 ) ∝ e − q ˜ 1 2 2 sinc 2 P 2 q ˜ 1 2 4 .
p F F S ( q ˜ 1 | q ˜ 2 = 0 ) ≈ 1 2 π e − q ˜ 1 2 2 .
H [ p F F S ( q ˜ 1 | q ˜ 2 = 0 ) ] ≈ 1 2 log 2 ( 2 π e ) .
p F F G ˜ ( q ˜ 1 | q ˜ 2 = 0 ) = 3 + P 2 6 π e − q ˜ 1 2 3 + P 2 6 .
N [ p F F S ( q ˜ 1 , q ˜ 2 ) ] = N [ p F F S ( q ˜ 1 | q ˜ 2 ) ] + N [ p F F S ( q ˜ 2 ) ] .
p N F S ( x ˜ 1 | x ˜ 2 = 0 ) ≈ 1 2 P A 1 sint 2 x ˜ 1 2 4 P 2 ,
( Δ x ˜ 1 | x ˜ 2 = 0 ) 2 = ∫ d x ˜ 1 x ˜ 1 2 p N F S ( x ˜ 1 | x ˜ 2 = 0 ) = 4 A 2 P 2 A 1 ,
p N F G ˜ ( x ˜ 1 | x ˜ 2 = 0 ) = A 1 2 2 π A 2 P e − A 1 8 A 2 P 2 x ˜ 1 2 .
N [ p N F S ( x ˜ 1 , x ˜ 2 ) ] = N [ p N F S ( x ˜ 1 | x ˜ 2 ) ] + N [ p N F S ( x ˜ 2 ) ] ,
n G T ≈ n G C + n G M ( iff   P ≪ 1 ) ,

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