## 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

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

## 1. Introduction

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

14. 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]

*K*, and the Schmidt modes allowed for each photon [15

15. 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]

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]

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

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

*simpler*EPR-criterion can still be used for the spatial entanglement detection. We also show that (and when) the EPR-criterion can be used as a witness for the non-Gaussianity of this state.

*negentropy*[26

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

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]

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]

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]

## 2. The phase-matching conditions and the variances of the conditional probabilities

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]

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

14. 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]

*q*_{1},

*q*_{2}〉 represents a two-photon state in plane-wave modes whose transverse wave vectors are

*q*_{1}and

*q*_{2}.

*L*is the crystal length,

*k*

*is the pump beam wave number,*

_{p}*w*

_{0}is the pump beam waist, which is located at

*z*= 0. This state may be rewritten in the coordinate space as [23

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]

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]

*x*) being the sine integral function. The functions sinc(

*b*|

**|**

*q*^{2}/2) and

*bx*

^{2}) by

*e*

^{−αbx2}. Sometimes it is used that

*α*= 1 [15

15. 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]

19. 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]

*α*is chosen such that both functions coincide at 1/

*e*

^{2}[17

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

*e*[22

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]

15. 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]

*α*

*represent distinct Gaussian approximations for the sinc function. In Fig. 1(a) [(b)] we compare the curves*

_{i}*x̃*

_{2},

*q̃*

_{2}= 0,

*σ*= 1 (and the crystal centered at

*z*= 0), for the case where the dimensionless parameter

*P*= 0.1. This parameter has been used in the study of the quantification of the spatial entanglement [15

**92**, 127903 (2004). [CrossRef] [PubMed]

16. 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]

*P*can be reached in three different ways. Here we considered the values of

*L*and

*k*

*as fixed parameters such that*

_{p}*P*varies with

*w*

_{0}. In this case we find

*α*

_{1}= 0.45 (

*α*

_{2}= 0.72) for the case where the sinc and Gaussian functions coincide at 1/

*e*(1/

*e*

^{2}). From Fig. 1(a)–1(b) one can see that for a small value of

*P*, the Gaussian approximation only describes properly the momentum conditional distribution. The position conditional distribution is barely described by the approximation. In Fig. 1(c) and Fig. 1(d) we have the same type of analysis but now for a larger value of

*P*. In this case, the Gaussian approximation is useful only for describing the position conditional distribution. The overall behavior of the Gaussian approximations is showed in Fig. 1(e) and Fig. 1(f), where the normalized variances of the far- [(Δ

*q*

_{1}|

_{q2})

^{2}

*L/k*

*] and near-field [(Δ*

_{p}*x*

_{1}|

_{x2})

^{2}

*k*

_{p}*/L*] conditional distributions are plotted in terms of

*P*.

## 3. The EPR-Criterion as a witness for the non-Gaussianity of the spatial two-photon state

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]

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

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]

*x*

_{1}|

_{x2})

^{2}and (Δ

*q*

_{1}|

_{q2})

^{2}, we can also look for its effect on the EPR-criterion. This is showed with the red (solid) line in Fig. 2, which was calculated for the values of

*x̃*

_{2}and

*q̃*

_{2}at the origin. For smaller values of

*P*, the conditional variances are independent of the

*x̃*

_{2}and

*q̃*

_{2}values [1

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]

*P*increases, the variances become dependent on

*x̃*

_{2}and

*q̃*

_{2}. Nevertheless for smaller values of

*x̃*

_{2}and

*q̃*

_{2}, which are of most experimental relevance, the red (solid) curve shown in Fig. 2 captures the overall behavior of the product of (Δ

*x*

_{1}|

_{x2})

^{2}and (Δ

*q*

_{1}|

_{q2})

^{2}for the state of Eq. (1). As we can see, for values of

*P*smaller than 0.56 or greater than 2.58, the EPR-criterion can safely be used for detecting the spatial entanglement of the two-photon state of SPDC.

## 4. Quantifying the non-Gaussianity of the spatial two-photon state of SPDC

*negentropy*which is the base of our approach [26

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

*p*(

*ξ*

_{1},

*ξ*

_{2}) is defined as

*N*≡

*H*[

*p*

*(*

^{G̃}*ξ*

_{1},

*ξ*

_{2})] –

*H*[

*p*(

*ξ*

_{1},

*ξ*

_{2})], where

*p*

^{G}^{̃}(

*ξ*

_{1},

*ξ*

_{2}) is a Gaussian distribution with the same expected values and covariance matrix of

*p*(

*ξ*

_{1},

*ξ*

_{2}). The function

*H*[

*p*(

*ξ*

_{1},

*ξ*

_{2})], called

*differential entropy*, is defined as

*H*[

*p*(

*ξ*

_{1},

*ξ*

_{2})] ≡ − ∫

*d*

*ξ*

_{1}

*d*

*ξ*

_{2}

*p*(

*ξ*

_{1},

*ξ*

_{2}) log

_{2}

*p*(

*ξ*

_{1},

*ξ*

_{2}) [32

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]

*i, j*= 1,2 and

*i*≠

*j*. According to these definitions, one can observe that the non-Gaussianity of the spatial state of SPDC decreases under partial trace, such that

*nG*

*>*

^{T}*nG*

*; and that it is additive when the composite system is represented by a product state, i.e., if |Ψ〉 is a product state, then*

^{M}*nG*

*= 2*

^{T}*nG*

*[see Appendix D]. These are common properties with the QRE measure of [27*

^{M}**78**, 060303 (2008). [CrossRef]

**82**, 052341 (2010). [CrossRef]

*P*. It is interesting to note that these curves quantify the idea already presented in Fig. 1(e)–1(f). As larger the value of

*P*is, the less the conditional momentum distribution can be approximated by a Gaussian function. On the other hand, the conditional position probabilities tends to a normal distribution when

*P*increases. In Fig. 3(b) we plot the negentropies of the near- and far-field marginal probabilities. One can see that they have a different dependence on

*P*in comparison with the conditional probabilities. Now, the near-field distribution tends to a Gaussian function for larger values of

*P*, and the far-field one for smaller values of

*P*. In Fig. 3(c) and Fig. 3(d) we have

*nG*

*and*

^{C}*nG*

*plotted in terms of*

^{M}*P*. The insets of these figures show the corresponding near- and far-field distributions at the points of minimum, indicated with red circles.

*P*≪ 1, one can use the relation between the joint and conditional density probabilities to decompose

*nG*

*as the sum of*

^{T}*nG*

*and*

^{C}*nG*

*: In a typical experimental configuration for SPDC, where the pump beam spot size is around 1 mm at the crystal plane, the value of*

^{M}*P*can be smaller than 0.05. Thus, in general, the total non-Gaussianity of the spatial two-photon state of SPDC can be estimated in terms of the (easier to measure) near- and far-field negentropies of the conditional and marginal distributions. This simplify the measurement of

*nG*

*, since there is no need to scan the whole transverse planes associated with the near- and far-field joint distributions.*

^{T}## 5. Conclusion

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

**78**, 060303 (2008). [CrossRef]

**82**, 052341 (2010). [CrossRef]

## A. EPR-criterion for spatial Gaussian two-photon states

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

14. 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]

*q*

_{1},

*q*

_{2}) is given by with Δ(

*q*

_{1}+

*q*

_{2}) being the angular spectrum of the pump beam and Θ(

*q*

_{1}–

*q*

_{2}) representing the phase-matching conditions of the non-linear process. If both functions are represented by Gaussian functions of the form

**92**, 127903 (2004). [CrossRef] [PubMed]

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

*δ*

_{−}can be adjusted in order to approximate the phase-matching function by distinct Gaussian functions. For a pump laser beam with a Gaussian transverse profile,

*c*is the radius of this pump at the plane of the non-linear crystal.

*x*

_{2}and

*q*

_{2}[1

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]

*x*

_{2}= 0 and

*q*

_{2}= 0. In this case

*P*(

*x*

_{1}|

*x*

_{2}= 0) and

*P*(

*q*

_{1}|

*q*

_{2}= 0) are given by and Here

**92**, 127903 (2004). [CrossRef] [PubMed]

*K*for Gaussian states. It was showed in Ref. [15

**92**, 127903 (2004). [CrossRef] [PubMed]

*x*

_{1}|

_{x2})

^{2}(Δ

*q*

_{1}|

_{q2})

^{2}is equal to

*K*= 1 (i.e., for product Gaussian states).

## B. Mancini *et al.* Criterion for the spatial entanglement of SPDC

*et al.*[31

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

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]

*q̃*

_{+}=

*q̃*

_{1}+

*q̃*

_{2}and

*x̃*

_{−}=

*x̃*

_{1}–

*x̃*

_{2}. The constants and

*P*, while considering the state of SPDC and when distinct Gaussian approximations for this state are considered. Since there is no upper limit for this criterion with Gaussian two-mode states, it is not possible to use the Mancini-Criterion for the detection of the non-Gaussianity of the spatial state of SPDC.

## C. Negentropy of far- and near-field joint distributions of SPDC

## C.1. Negentropy of a probability density function

*p*(

*ξ*

_{1},

*ξ*

_{2}) be a probability density function. The Negentropy (

*N*) of

*p*(

*ξ*

_{1},

*ξ*

_{2}) is defined as [26

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

*p*

*(*

^{G̃}*ξ*

_{1},

*ξ*

_{2}) is a Gaussian distribution with the same expected value vector

**= {**

*μ**μ*

_{ξ1},

*μ*

_{ξ2}} and same covariance matrix Λ of

*p*(

*ξ*

_{1},

*ξ*

_{2}). The function

*H*[

*p*(

*ξ*

_{1},

*ξ*

_{2})], called

*differential entropy*, is defined as [32

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

## C.2. Negentropy of far-field joint distribution

*μ**= {〈*

_{FF}*q̃*

_{1}〉, 〈

*q̃*

_{2}〉} are null. So, we have that and

## C.3. Negentropy of near-field joint distribution

*C*is a normalization constant given by

*I*≈ −0.692. Then, the differential entropy of the near-field joint distribution is

## D. Further properties of *nG*^{T}

^{T}

**Property 1.**

*nG*

*= 0*

^{T}*iff the spatial two-photon state of SPDC is a Gaussian state.*

*Proof.*Since and that negentropy is always nonnegative, we have that

*H*is a continuous and monotonic function, it holds that and

**Property 2.**

*If the spatial two-photon state is a product state then nG*

^{T}*is additive, i.e., nG*

*= 2 ×*

^{T}*nG*

*.*

^{M}*Proof.*For a set of two random and independent variables, the joint probability density functions are given by the product of the probability density functions associated to each variable, i.e.,

*p*(

*ξ*

_{1},

*ξ*

_{2}) =

*p*(

*ξ*

_{1})

*p*(

*ξ*

_{2}). Such type of probability density function has the covariance matrix Λ

*defined by the elements and a expected value vector*

_{ij}**= {〈**

*μ**ξ*

_{1}〉, 〈

*ξ*

_{2}〉}.

*p*

*(*

^{G̃}*ξ*

_{1},

*ξ*

_{2}) with the same covariance matrix and expected value vector is given by

**= {**

*ξ**ξ*

_{1},

*ξ*

_{2}}.

*p*(

*ξ*

_{1}) and

*p*(

*ξ*

_{2}). Since the random variables are independent, it holds that and consequently, we have that the negentropy of

*p*(

*ξ*

_{1},

*ξ*

_{2}) can be written as

*nG*

^{M1}=

*nG*

^{M2}=

*nG*

*. Then which proves our statement.*

^{M}**Property 3.**

*The non-Gaussianity of the spatial state of SPDC decreases under partial trace, such that nG*

*>*

^{T}*nG*

*.*

^{M}*Proof.*To demonstrate this we first study the negentropies of the far- and near-field marginal distributions, while considering two limiting cases: (i)

*P*≪ 1 and (ii)

*P*very large. The joint distributions of the far- and near-field planes are given by Eq. (27) and Eq. (40), respectively. Thus, the marginal distributions are given by and

^{2}(

*P*

^{2}

*q̃*

_{2}). It gives that Then, for values of

*P*≪ 1,

*nG*

*≈ 0.154 such that*

^{M}*nG*

*>*

^{T}*nG*

*.*

^{M}*P*,

*nG*

*≈ 0.224 which is also smaller than*

^{M}*nG*

*.*

^{T}*P*< 3, have been calculated numerically. The values of the negentropies of the far- and near-field marginal distributions are shown in Fig. 3(b) of the main paper. In Fig. 3(d),

*nG*

*is plotted in terms of*

^{M}*P*. As one can see, in such intermediate cases, one always have that

*nG*

*>*

^{T}*nG*

*. For*

^{M}*P*= 3 the value of

*nG*

*≈ 0.175. For values of*

^{M}*P*> 9,

*nG*

*already tends to its maximal value which is*

^{M}## E. Calculating the non-Gaussianity of the state of the spatially correlated down-converted photons using the QRE

*δ*

*[27*

_{B}**78**, 060303 (2008). [CrossRef]

**82**, 052341 (2010). [CrossRef]

*p*(

*x*

*,*

_{i}*q*

*) (where*

_{j}*i*,

*j*= 1, 2 and

*i*≠

*j*) are described by a product of the marginal position and momentum distributions

*p*(

*x*

*) and*

_{i}*p*(

*q*

*). In such case the covariance matrix can be written, considering the order*

_{j}**V**(

*x̃*

_{1},

*q̃*

_{1},

*x̃*

_{2},

*q̃*

_{2}), as

*μ*= 0.44 [28

**82**, 052341 (2010). [CrossRef]

*S*= 1.08. Since the state of Eq. (1) is pure, we have that

*δ*

*= 1.08, which does not depend on the value of*

_{B}*P*.

## F. nG^{T} at the limit when P ≪ 1

*nG*

*when considering the limit of*

^{T}*P*≪ 1, and that one of the down-converted photon is detected in transverse points around the origin. First we analyze the negentropy of far-field joint distribution: In the limits considered, the differential entropies can be written as

*H*[

*p*(

*ξ*

_{1},

*ξ*

_{2})] =

*H*[

*p*(

*ξ*

_{1}|

*ξ*

_{2})] +

*H*[

*p*(

*ξ*

_{2})], such that the negentropy of the far-field joint distribution may also be rewritten as

*q̃*

_{2}= 0 will be given by When

*P*≪ 1, the sinc function is much larger than the Gaussian function, such that we can approximate this distribution as Therefore, the differential entropy is Note that Eq. (88) is a normal distribution with the expected value equal to zero and the variance equal to 1.

*q̃*

_{2}= 0 equal to

*P*≪ 1, this Gaussian distribution has also the expected value equal to zero and the variance equal to 1. Thus, it corresponds to the Gaussian distribution with the same expected value and variance of

*in this limit*, we can define the negentropy of the far-field conditional distribution in terms of the

*P*≪ 1, then the near-field conditional distribution is given by where we assumed

*σ*= 1 for simplicity.

*A*

_{1}and

*A*

_{2}are the constants defined above. Now we consider the near-field Gaussian joint distribution

*P*≪ 1, is given by

*in this limit*we can write the negentropy of the near-field joint distribution as where

## Acknowledgments

## References and links

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. |

2. | M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. |

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

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. |

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. |

6. | S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. |

7. | 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 |

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. |

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. |

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

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. |

12. | C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A |

13. | 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 |

14. | 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 |

15. | C. K. Law and J. H. Eberly, “Analysis and Interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett. |

16. | 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 |

17. | K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A |

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 |

19. | 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 |

20. | D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A |

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 |

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 |

23. | H. Di Lorenzo Pires and M. P. van Exter, “Observation of near-field correlations in spontaneous parametric down-conversion,” Phys. Rev. A |

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. |

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. |

26. | A. Hyvärinen, J. Karhunen, and E. Oja, |

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 |

28. | M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A |

29. | M. Ostermeyer, D. Korn, D. Puhlmann, C. Henkel, and J. Eisert, “Two-dimensional characterization of spatially entangled photon pairs,” J. Mod. Opt. |

30. | H. Di Lorenzo Pires and M. P. van Exter, “Near-field correlations in the two-photon field,” Phys. Rev. A |

31. | S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. |

32. | T. M. Cover and J. A. Thomas, |

33. | P. Comon, “Independent component analysis, A new concept?,” Sig. Process. |

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. |

**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.