## Quantifying the non-Gaussianity of the state of spatially correlated down-converted photons |

Optics Express, Vol. 20, Issue 4, pp. 3753-3772 (2012)

http://dx.doi.org/10.1364/OE.20.003753

Enhanced HTML Acrobat PDF (1108 KB)

### 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 OSA

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: December 12, 2011

Revised Manuscript: January 22, 2012

Manuscript Accepted: January 23, 2012

Published: January 31, 2012

**Citation**

E. S. Gómez, W. A. T. Nogueira, C. H. Monken, and G. Lima, "Quantifying the non-Gaussianity of the state of spatially correlated down-converted photons," Opt. Express **20**, 3753-3772 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3753

Sort: Year | Journal | Reset

### References

- 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]
- A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev.47, 777 (1935). [CrossRef]
- 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]
- 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]
- 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]
- T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A52, R3429 (1995). [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]
- C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A31, 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. A505122 (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. A57, 3123 (1998). [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]
- H. Di Lorenzo Pires, C. H. Monken, and M. P. van Exter, “Direct measurement of transverse-mode entanglement in two-photon states,” Phys. Rev. A80, 022307 (2009). [CrossRef]
- K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A75, 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. A83, 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. A76, 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. A78, 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. A79, 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. B42, 114011 (2009). [CrossRef]
- H. Di Lorenzo Pires and M. P. van Exter, “Observation of near-field correlations in spontaneous parametric down-conversion,” Phys. Rev. A79, 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]
- A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001). [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. A78, 060303 (2008). [CrossRef]
- M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A82, 052341 (2010). [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. A80, 053820 (2009). [CrossRef]
- S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett.88, 120401 (2002). [CrossRef] [PubMed]
- T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991). [CrossRef]
- P. Comon, “Independent component analysis, A new concept?,” Sig. Process.36, 287–314 (1994). [CrossRef]
- 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].
- M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett.96, 080502 (2006). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.