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

The state of the 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 autocorrelation covariance sub-matrices, and will be relevant for new applications of the spatial correlation of SPDC in CV quantum information processing.


Introduction
In spontaneous parametric down-conversion (SPDC), photon pairs are generated with several degrees of freedom quantum correlated.In particular, the down-converted photons are spatially entangled.Due to energy and momentum conservation, the sum of the transverse momenta and the difference of the transverse positions of the photons can be well defined, even though the position and momentum of each photon are undefined [1,2].This type of Einstein-Podolsky-Rosen (EPR) correlation [3], has been used as a resource for fundamental studies of quantum mechanics [1,2,[4][5][6], for quantum imaging [7,8], and experiments of quantum information [9][10][11].
The process of SPDC has been studied extensively in the past [12][13][14], and much of the recent effort has been put in the quantification of the spatial entanglement for a given experimental geometry.Traditionally, this has been done through the technique of Schmidt decomposition of the two-photon wave function, which gives the Schmidt number, K, and the Schmidt modes allowed for each photon [15][16][17][18].While this approach describes important properties of the spatial correlation, it basically gives no information of the form of the spatial joint distributions, therefore giving no information about the Gaussianity of the spatial two-photon state of SPDC.
Moreover, it is well known that the state of the down-converted photons depends on the phase-matching conditions.Nevertheless, due to its complex structure, the phase-matching functions are usually approximated by Gaussian functions [17][18][19][20][21][22].While the approach being adopted seems to be reasonable, it has already been shown that fine (new) structures in the spatial distributions of these photons can be observed due to (the manipulation of) the phasematching conditions [23].
In this work we study the non-Gaussianity of the spatial two-photon state of SPDC by properly taking into account the phase-matching conditions.We start by showing how the variances of the near-and far-field conditional probability distributions are affected by the phasematching functions.Then, we analyze the role of the EPR-criterion [3,24] regarding the non-Gaussianity and entanglement detection of this state.Even though it has been demonstrated that higher order separability criteria can be used for the entanglement detection of spatial non-Gaussian entangled states [25], we show that a proper consideration of the phase-matching function reveals, precisely, when the 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.
Furthermore, we introduce a statistical measure, based on the negentropy [26] of the nearand far-field joint distributions, which allows for the quantification of the non-Gaussianity of the spatial two-photon state of SPDC.This measure does not correspond to a quantum mechanical generalization of the negentropy, such as the non-Gaussianity measure based on the quantum relative entropy (QRE) [27,28], and so does not require the knowledge of the full density matrix.Only the moments associated with the diagonal sub-matrices of the covariance matrix need to be measured.Thus, it is experimentally more accessible [29].Moreover, for most of the configurations used so far, we show that its value can be estimated from the (easier to measure) marginal and conditional distributions.We also demonstrate that it has common properties with previous introduced measures of non-Gaussianity [27,28].The quantification of the non-Gaussianity of a quantum state has important applications for quantum information [25,27,28], and thus the practicality our measure shall be relevant for new applications of the spatial correlations of SPDC in this field.

The phase-matching conditions and the variances of the conditional probabilities
We consider the process of quasi-monochromatic SPDC, in the paraxial regime, for configurations with negligible Poynting vector transverse walk-off, that can be obtained using noncritical phase-matching techniques [23].In the momentum representation, the two-photon state is given by [12][13][14] |Ψ ∝ dq q q 1 dq q q 2 e − ζ 4 |q q q 1 +q q q 2 | 2 sinc L|q q q 1 − q q q 2 | 2 4k p | q q q 1 , q q q 2 , ( where | q q q 1 , q q q 2 represents a two-photon state in plane-wave modes whose transverse wave vectors are q q q 1 and q q q 2 .L is the crystal length, k p is the pump beam wave number, ζ = w 2 0 − 2iz/k p , and w 0 is the pump beam waist, which is located at z = 0.This state may be rewritten in the coordinate space as [23,30] where we define the function sint as sint (x) ≡ 2 π ∞ x dt sinct ≡ 1 − 2 π Si (x), Si (x) being the sine integral function.The functions sinc (b |q q q| 2 /2) and π 2b sint (|ρ ρ ρ| 2 /2b) form a Fourier transform pair.
The sinc and sint functions arise from the phase-matching conditions, and due to the difficulty of dealing analytically with them, they are usually approximated by Gaussian functions.The approach that has been adopted consists in approximating the function sinc (bx 2 ) by e −αbx 2 .Sometimes it is used that α = 1 [15,19], and in other cases the value of α is chosen such that both functions coincide at 1/e 2 [17,20,21] or at 1/e [22] of their peak.While this approximation seems to be reasonable for the entanglement quantification [15], there has been no investigation to determine how precise it is for describing the distribution of the momentum correlations.Besides, it should be noticed that once a Gaussian approximation is adopted for the sinc function, the corresponding approximation for the sint function is already defined by the Fourier transform.Therefore, it is also not clear that such approximation is indeed good for describing the position distributions.Thus, it is not clear whether the SPDC two-photon state can indeed be written as a two-mode Gaussian state.
To investigate this point we study how the variances of the momentum (far-field) and position (near-field) conditional probability distributions are affected by the phase-matching function, and compare the obtained results with the cases where Gaussian approximations are considered.Due to the symmetry of the two-photon wave functions, there is no loss of generalization if we work in one dimension (i.e., y 1 = y 2 = 0 and q y1 = q y2 = 0).To simplify our analysis we define the following dimensionless variables: x j = x j /w 0 and q j = w 0 q j , j = 1, 2. The probabilities for coincidence detection at the far-and near-field planes are where σ 2 = (z 2 0 + z 2 )/z 2 0 , P = L/(2z 0 ), and z 0 = k p w 2 0 /2 is the diffraction length of the pump beam.The joint probabilities are related with the conditional (and marginal) probabil- . In the case of the Gaussian approximations discussed above, the probabilities of coincidence detection are where different values of α i represent distinct Gaussian approximations for the sinc function.In Figure 1(a) [(b)] we compare the curves p S FF and p G FF,α i (p S NF and p G NF,α i ) considering x2 , q2 = 0, σ = 1 (and the crystal centered at z = 0), for the case where the dimensionless parameter Fig. 1.The far-and near-field conditional probability distributions, while considering the state of SPDC (p S FF and p S NF ) and when some Gaussian approximations are assumed (p G FF,α i and p G NF,α i ).In (a)-(d) the curves p S FF and p S NF (red solid lines) are compared with p G FF,α 2 and p G NF,α 2 for α 1 = 0.45 (green dashed line), α 2 = 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.α 3 = 1.P = 0.1.This parameter has been used in the study of the quantification of the spatial entanglement [15,16], and it brings universality to the theory since a certain value of P can be reached in three different ways.Here we considered the values of L and k p as fixed parameters such that 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 Figure 1(a)-(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 Figure 1(c) and Figure 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 Figure 1(e) and Figure 1(f), where the normalized variances of the far-[(∆q 1 | q 2 ) 2 L/k p ] and near-field [(∆x 1 | x 2 ) 2 k p /L] conditional distributions are plotted in terms of P.

The EPR-Criterion as a witness for the non-Gaussianity of the spatial two-photon state
The near and far-field conditional probabilities can be used for implementing the EPR-paradox [1,3,21,24].This is done by observing the violation of the inequality , which certifies the quantum nature of the spatial correlations of the down-converted photons.Since we have determined how the phase-matching function affects the variances (∆x 1 | x 2 ) 2 and (∆q 1 | q 2 ) 2 , we can also look for its effect on the EPR-criterion.This is showed with the red (solid) line in Figure 2, which was calculated for the values of x2 and q2 at the origin.For smaller values of P, the conditional variances are independent of the x2 and q2 values [1].Whenever P increases, the variances become dependent on x2 and q2 .Nevertheless for smaller values of x2 and q2 , which are of most experimental relevance, the red (solid) curve shown in Figure 2 captures the overall behavior of the product of (∆x 1 | x 2 ) 2 and (∆q 1 | q 2 ) 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.
Besides of being useful as a entanglement witness, the EPR-criterion can also be used as a witness for the non-Gaussianity of the spatial state of SPDC.This emphasizes another application for this criterion, which has been related already with other quantum information tasks [24].To observe this, we note that for a pure two-mode Gaussian state it is possible to show that (∆x such that 1 4 is an upper bound for the EPR-criterion with these states.Since it has been demonstrated in [15] that the two-photon state of Eq. ( 1) is always entangled, we can say whenever the product of variances is greater than 1 4 , that it witnesses the non-Gaussianity of the entangled spatial two-photon state of SPDC.As one can see in Figure 2, this happens for 0.56 ≤ P ≤ 2.58.The Schmidt decomposition of the state wave function, used together with the EPR-criterion value, reveals the non-Gaussianity of a two-mode entangled state.Such observation does not necessarily hold true when other second-order moments criteria are considered.This is shown in Appendix B for the criterion of Ref. [31], and for the states considered in Figure 2.  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 α 1 , α 2 and α 3 , respectively.

Quantifying the non-Gaussianity of the spatial two-photon state of SPDC
From our previous analysis it is clear that the spatial state of SPDC can not be seen as a twomode Gaussian entangled state, even when it is generated with a Gaussian pump beam and in the case of perfect phase-matching.We now proceed to quantifying the non-Gaussianity of this state.First, we introduce the concept of negentropy which is the base of our approach [26].The negentropy of a probability density function p(ξ 1 , ξ 2 ) is defined as 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 [32].The advantage of using negentropy is that it can be seen as the optimal estimator of non-Gaussianity, as far as density probabilities are involved.This is due to the properties that it is always non-negative, and that it is zero only for Gaussian distributions.Besides, it is invariant under invertible linear transformations [33].
Motivated by these properties we define the total non-Gaussianity of the spatial two-photon state of SPDC as where )] ≈ 0.22.Thus, the total non-Gaussianity of the spatial two-photon state of SPDC [Eq (1)] is nG T ≈ 0.37.It is interesting to note that it does not depend on P.This was expected since the phase-matching functions do not change their functional form when P varies.The calculations performed for these negentropies can be adapted for different experimental geometries, or used for the proper determination of other quantum information quantities related with the differential entropy of the spatial joint distributions [34].The fact that nG T = 0, emphasizes that spatial Gaussian approximations should be taken carefully due to the extremality of Gaussian states [35].For comparison purposes, we calculated in Appendix E the value of δ B , which is the measure of non-Gaussianity based on the QRE [27, 28].We obtain that δ B = 1.08 and such result also highlight the non-Gaussian character of the state of Eq. ( 1).Furthermore, it has the same behavior of nG T , since it does not depend on the parameter P.
In analogy with Eq. ( 7), we define the non-Gaussianity of the conditional and marginal distributions as: nG C ≡ N[p S FF (q i |q j )] + N[p S NF (x i |x j )] and nG M ≡ N[p S FF (q i )] + N[p S NF (x i )] with 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 M ; and that it is additive when the composite system is represented by a product state, i.e., if | Ψ is a product state, then nG T = 2nG M [see Appendix D].These are common properties with the QRE measure of [27,28].
In Figure 3(a) we plot the negentropies N[p S FF (q 1 |q 2 )] and N[p S NF (x 1 |x 2 )] as a function of P. It is interesting to note that these curves quantify the idea already presented in Figure 1(e)-(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 Figure 3(b) we plot the negentropies of the nearand 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 Figure 3(c In a typical experimental configuration for SPDC, where the pump beam spot size is around 1 mm at the crystal plane, the value of 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 T , since there is no need to scan the whole transverse planes associated with the near-and far-field joint distributions.

Conclusion
We have investigated the spatial distributions of the entangled down-converted photons by proper considering the phase-matching conditions.By understanding how the near-and farfield conditional distributions are affected by the phase-matching function, we could show that the EPR-criterion [3] can be used as a witness for the non-Gaussianity of the spatial state of the SPDC.The work culminated in the quantification of the non-Gaussianity of this state, which was based in a new and very experimentally accessible measure.As it has been discussed in [27,28], the quantification of the non-Gaussianity of a quantum state has many applications in the area of continuous variables quantum information processing.Thus, we envisage the use of our result for new applications of the spatial correlations of SPDC.

A. EPR-criterion for spatial Gaussian two-photon states
As we mentioned in the main paper, the momentum representation of the spatial two-photon state generated in the spontaneous parametric down-conversion (SPDC) process, under paraxial approximation and for configurations with negligible Poynting vector transverse walk-off, can be written as [12][13][14] where the amplitude Ψ(q 1 , q 2 ) is given by with ∆(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 ∆(q [17][18][19][20][21][22], we have in momentum representation that and in position representation that The parameter 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, σ 2 + = 2/c 2 , where c is the radius of this pump at the plane of the non-linear crystal. Based on Eqs. ( 11) and ( 12), we can obtain the probability density functions for the conditional position and momentum distributions of the down-converted photons.The variances of these curves are, in general, independent of the value considered for x 2 and q 2 [1] and here we use, for simplicity, 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 σ 2 + and δ 2 − are the widths of the respective Gaussian functions [15].The variances of the conditional distributions can be calculated directly from Eqs. ( 13) and ( 14), which results in and Then, the product of them is given by Let us now consider the Schmidt number K for Gaussian states.It was showed in Ref. [15] that it is given by the expression Comparing Eqs. ( 17) and ( 18), it follows immediately that the product of the conditional variances is a function of the Schmidt number, such that This expression is valid for any Gaussian approximation taken for the phase-matching function.It is possible to see that the maximal value of (∆x 1 | x 2 ) 2 (∆q 1 | q 2 ) 2 is equal to 1  4 and that it happens when the Schmidt number K = 1 (i.e., for product Gaussian states).

B. Mancini et al. Criterion for the spatial entanglement of SPDC
Another entanglement detection criterion, based on second-order moments, is the one introduced by Mancini et al. [31].For the case of spatial entanglement it reads [29] When this inequality is violated, the state is not separable.If this condition is satisfied, no information can be drawn.For the far and near-field joint spatial distributions [Eq.( 3) and Eq. ( 4) of main paper, respectively], the variances in (20) are given by where q+ = q1 + q2 and x− = x1 − x2 .The constants and Figure 4 shows the dependence of the Mancini-Criterion with the parameter 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.2. Negentropy of far-field joint distribution
The joint probability density function in the far-field plane is given by The covariance matrix of p S FF ( q1 , q2 ) is given by The elements in the diagonal are the variances of the marginal distributions p S FF ( q1 ) and p S FF ( q2 ).In order to calculate the terms of Λ S FF , we first note that the expected values µ µ µ FF = { q1 , q2 } are null.So, we have that and Let us consider the following Gaussian approximation for describing the joint probability density distribution at the far-field plane: The elements of the covariance matrix Λ G FF of p G FF ( q1 , q2 ) are given by and From Eq. ( 29), ( 30), (32) and (33) we can observe that the condition implies that the joint distributions ( 27) and ( 31) have the same covariance matrix.Hereafter, the Gaussian distribution with the same covariance matrix of ( 27) is denoted by p G FF ( q1 , q2 ).It is possible to show that the differential entropy of the Gaussian distribution p G FF ( q1 , q2 ) is The differential entropy of p S FF ( q1 , q2 ) is given by Making u = q1 + q2 √ 2 and v = P 2 ( q1 − q2 ) we obtain that The last integral in (37) must be calculated numerically and it gives that Therefore, the differential entropy of p S FF ( q1 , q2 ) is According to Eq. ( 25), we have that − 1.17

C.3. Negentropy of near-field joint distribution
The joint probability density function in the near-field plane is given by where C is a normalization constant given by The covariance matrix for the near-field joint distribution is where and Let us now consider the following Gaussian approximation for the joint probability distribution at the near-field plane: The covariance matrix where and From Eq. ( 43), ( 44), ( 47) and (48) we can observe that the condition must be satisfied for having Λ G NF = Λ S NF .The Gaussian distribution which satisfies this condition is denoted by p G NF ( x1 , x2 ).The differential entropy of this Gaussian distribution is given by The differential entropy for p S NF ( x1 , x2 ) reads The integral I = ds sint 2 s 2 ln sint 2 s 2 , must be calculated numerically.It gives that I ≈ −0.692.Then, the differential entropy of the near-field joint distribution is Therefore, the negentropy of the near-field joint distribution is and Since the differential entropy H is a continuous and monotonic function, it holds that Thus, we conclude that the two-photon state is a Gaussian state.
In the opposite direction we consider that the two-photon state is a Gaussian state.Then, the joint probability density functions of transverse position and momentum are Gaussian functions.So, from the definition of negentropy we must have that 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 i j defined by the elements The Gaussian distribution p G(ξ 1 , ξ 2 ) with the same covariance matrix and expected value vector is given by where ξ ξ ξ = {ξ 1 , ξ 2 }.Note that these marginal Gaussian distributions have the same variance and expected value that p(ξ 1 ) and p(ξ 2 ).Since the random variables are independent, it holds that H[p(ξ 1 , ξ 2 )] = H[p(ξ 1 )] + H[p(ξ 2 )], (70) and consequently, we have that the negentropy of p(ξ 1 , ξ 2 ) can be written as

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 α 1 , α 2 and α 3 , respectively.

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.