1. 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, 4–6], for quantum imaging [7, 8], and experiments of quantum information [9–11].

The process of SPDC has been studied extensively in the past [12–14], 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 [15–18]. 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 [17–22]. 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 phase-matching 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 phase-matching 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 near-and 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.

2. 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 [12–14]

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(bx²) by e^−αbx2. 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/e² [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., y₁ = y₂ = 0 and q_y1 = q_y2 = 0). To simplify our analysis we define the following dimensionless variables: x̃_j = x_j/w₀ and q̃_j = w₀q_j, j = 1, 2. The probabilities for coincidence detection at the far- and near-field planes are

In the case of the Gaussian approximations discussed above, the probabilities of coincidence detection are

 

Fig. 1 The far- and near-field conditional probability distributions, while considering the state of SPDC ( $p_{FF}^S$ and $p_{NF}^S$) and when some Gaussian approximations are assumed ( $p_{FF,{\alpha }_i}^G$ and $p_{NF,{\alpha }_i}^G$). In (a)–(d) the curves $p_{FF}^S$ and $p_{NF}^S$ (red solid lines) are compared with $p_{FF,{\alpha }_2}^G$ and $p_{NF,{\alpha }_2}^G$ 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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3. 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 ${({\mathrm{\Delta }{x}_1|}_{x_2})}^2{({\mathrm{\Delta }{q}_1|}_{q_2})}^2\ge \frac{1}{4}$, 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 (Δx₁|_x2)² and (Δq₁|_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 x̃₂ and q̃₂ at the origin. For smaller values of P, the conditional variances are independent of the x̃₂ and q̃₂ values [1]. Whenever P increases, the variances become dependent on x̃₂ and q̃₂. Nevertheless for smaller values of x̃₂ and q̃₂, which are of most experimental relevance, the red (solid) curve shown in Fig. 2 captures the overall behavior of the product of (Δx₁|_x2)² and (Δq₁|_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.

 

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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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 ${\left({\mathrm{\Delta }{x}_1|}_{x_2}\right)}^2{\left({\mathrm{\Delta }{q}_1|}_{q_2}\right)}^2=\frac{1}{4K}$ [see Appendix A], such that $\frac{1}{4}$ 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 $\frac{1}{4}$, that it witnesses the non-Gaussianity of the entangled spatial two-photon state of SPDC. As one can see in Fig. 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 Fig. 2.

4. 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 two-mode 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(ξ₁, ξ₂) is defined as N ≡ H[p^G̃(ξ₁, ξ₂)] – H[p(ξ₁, ξ₂)], where p^G̃(ξ₁, ξ₂) is a Gaussian distribution with the same expected values and covariance matrix of p(ξ₁, ξ₂). The function H[p(ξ₁, ξ₂)], called differential entropy, is defined as H[p(ξ₁, ξ₂)] ≡ − ∫dξ₁dξ₂ p(ξ₁, ξ₂) log₂ p(ξ₁, ξ₂) [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

In analogy with Eq. (7), we define the non-Gaussianity of the conditional and marginal distributions as: $n{G}^C\equiv N\left[p_{FF}^S\left(q_i|q_j\right)\right]+N\left[p_{NF}^S\left(x_i|x_j\right)\right]$ and $n{G}^M\equiv N\left[p_{FF}^S\left(q_i\right)\right]+N\left[p_{NF}^S\left(x_i\right)\right]$ 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 Fig. 3(a) we plot the negentropies $N\left[p_{FF}^S\left(q_1|q_2\right)\right]$ and $N\left[p_{NF}^S\left(x_1|x_2\right)\right]$ as a function of 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^C and nG^M plotted in terms of P. The insets of these figures show the corresponding near- and far-field distributions at the points of minimum, indicated with red circles.

 

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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As it is shown in Appendix F [See also Fig. 3(c) and Fig. 3(d)], in the limit of P ≪ 1, one can use the relation between the joint and conditional density probabilities to decompose nG^T as the sum of nG^C and nG^M:

5. 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 far-field 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 [12–14]

(9)$$|\mathrm{\Psi }〉=\int d{q}_1\int d{q}_2\mathrm{\Psi }\left(q_1,q_2\right)|1q_1〉|1q_2〉,$$

where the amplitude Ψ(q₁, q₂) is given by

(10)$$\mathrm{\Psi }\left(q_1,q_2\right)\propto \mathrm{\Delta }\left(q_1+q_2\right)\mathrm{\Theta }\left(q_1-q_2\right),$$

with Δ(q₁ + q₂) being the angular spectrum of the pump beam and Θ(q₁ – q₂) representing the phase-matching conditions of the non-linear process. If both functions are represented by Gaussian functions of the form $\mathrm{\Delta }\left(q_1+q_2\right)=\text{exp}\left[-\frac{{\left(q_1+q_2\right)}^2}{2{\sigma }_{+}^2}\right]$ and $\mathrm{\Theta }\left(q_1-q_2\right)=\text{exp}\left[-\frac{{\left(q_1-q_2\right)}^2}{2{\delta }_{-}^2}\right]$ [15, 17–22], we have in momentum representation that

(11)$$\mathrm{\Psi }\left(q_1,q_2\right)\propto e^{-{\left(q_1+q_2\right)}^2/2{\sigma }_{+}^2}{e}^{-{\left(q_1-q_2\right)}^2/2{\delta }_{-}^2},$$

and in position representation that

(12)$$\mathrm{\Psi }\left(x_1,x_2\right)\propto e^{-{\sigma }_{+}^2{\left(x_1+x_2\right)}^2/8}{e}^{-{\delta }_{-}^2{\left(x_1-x_2\right)}^2/8}.$$

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, ${\sigma }_{+}^2=2/c^2$, 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 x₂ and q₂ [1] and here we use, for simplicity, x₂ = 0 and q₂ = 0. In this case P(x₁|x₂ = 0) and P(q₁|q₂ = 0) are given by

(13)$$P\left(x_1|x_2=0\right)=\frac{\sqrt{{\sigma }_{+}^2+{\delta }_{-}^2}}{2\sqrt{\pi }}\hspace{0.17em}\text{exp}\hspace{0.17em}\left[-\frac{{\sigma }_{+}^2+{\delta }_{-}^2}{4}{x}_1^2\right],$$

and

(14)$$P\left(q_1|q_2=0\right)=\frac{\sqrt{{\sigma }_{+}^2+{\delta }_{-}^2}}{\sqrt{\pi }{\sigma }_{+}^2{\delta }_{-}^2}\hspace{0.17em}\text{exp}\hspace{0.17em}\left[-\frac{{\sigma }_{+}^2+{\delta }_{-}^2}{{\sigma }_{+}^2{\delta }_{-}^2}{q}_1^2\right].$$

Here ${\sigma }_{+}^2$ and ${\delta }_{-}^2$ 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

(15)$${\left({\mathrm{\Delta }{x}_1|}_{x_2}\right)}^2=\frac{2}{{\sigma }_{+}^2+{\delta }_{-}^2},$$

and

(16)$${\left({\mathrm{\Delta }{q}_1|}_{q_2}\right)}^2=\frac{{\sigma }_{+}^2{\delta }_{-}^2}{2\left({\sigma }_{+}^2+{\delta }_{-}^2\right)}.$$

Then, the product of them is given by

(17)$${\left({\mathrm{\Delta }{x}_1|}_{x_2}\right)}^2{\left({\mathrm{\Delta }{q}_1|}_{q_2}\right)}^2=\frac{{\sigma }_{+}^2{\delta }_{-}^2}{{\left({\sigma }_{+}^2+{\delta }_{-}^2\right)}^2}.$$

Let us now consider the Schmidt number K for Gaussian states. It was showed in Ref. [15] that it is given by the expression

(18)$$K=\frac{1}{4}{\left(\frac{{\sigma }_{+}}{{\delta }_{-}}+\frac{{\delta }_{-}}{{\sigma }_{+}}\right)}^2=\frac{1}{4}\frac{{\left({\sigma }_{+}^2+{\delta }_{-}^2\right)}^2}{{\sigma }_{+}^2{\delta }_{-}^2}.$$

Comparing Eqs. (17) and (18), it follows immediately that the product of the conditional variances is a function of the Schmidt number, such that

(19)$${\left({\mathrm{\Delta }{x}_1|}_{x_2}\right)}^2{\left({\mathrm{\Delta }{q}_1|}_{q_2}\right)}^2=\frac{1}{4K}.$$

This expression is valid for any Gaussian approximation taken for the phase-matching function. It is possible to see that the maximal value of (Δx₁|_x2)²(Δq₁|_q2)² is equal to $\frac{1}{4}$ 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]

(20)$${\left[\mathrm{\Delta }\left({\tilde{x}}_1-{\tilde{x}}_2\right)\right]}^2{\left[\mathrm{\Delta }\left({\tilde{q}}_1+{\tilde{q}}_2\right)\right]}^2\ge 1.$$

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 Eq. (20) are given by

(21)$${\left[\mathrm{\Delta }\left({\tilde{q}}_1+{\tilde{q}}_2\right)\right]}^2\propto \int d{\tilde{q}}_{+}{\tilde{q}}_{+}^2{e}^{-\frac{1}{2}{\tilde{q}}_{+}^2}=1,$$

(22)$${\left[\mathrm{\Delta }\left({\tilde{x}}_1-{\tilde{x}}_2\right)\right]}^2\propto \int d{\tilde{x}}_{-}{\tilde{x}}_{-}^2{\text{sint}}^2\frac{1}{4P}{\tilde{x}}_{-}^2=4\frac{A_2}{A_1}{P}^2,$$

where q̃₊ = q̃₁ + q̃₂ and x̃₋ = x̃₁ – x̃₂. The constants

(23)$$A_1={\int }_{-\infty }^{\infty }d\xi \hspace{0.17em}{\text{sint}}^2{\xi }^2\approx 1.4008,$$

and

(24)$$A_2={\int }_{-\infty }^{\infty }d\xi \hspace{0.17em}{\xi }^2\hspace{0.17em}{\text{sint}}^2{\xi }^2\approx \mathrm{0.5897.}$$

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.

 

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

C.1. Negentropy of a probability density function

Let p(ξ₁, ξ₂) be a probability density function. The Negentropy (N) of p(ξ₁, ξ₂) is defined as [26]

(25)$$N\equiv H\left[p^{\tilde{G}}\left({\xi }_1,{\xi }_2\right)\right]-H\left[p\left({\xi }_1,{\xi }_2\right)\right],$$

where p^G̃(ξ₁, ξ₂) is a Gaussian distribution with the same expected value vector μ = {μ_ξ1, μ_ξ2} and same covariance matrix Λ of p(ξ₁, ξ₂). The function H[p(ξ₁, ξ₂)], called differential entropy, is defined as [32]

(26)$$H\left[p\left({\xi }_1,{\xi }_2\right)\right]\equiv -\int d{\xi }_1\int d{\xi }_{2}p\left({\xi }_1,{\xi }_2\right){\text{log}}_2\left(p\left({\xi }_1,{\xi }_2\right)\right).$$

C.2. Negentropy of far-field joint distribution

The joint probability density function in the far-field plane is given by

(27)$$p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)=\frac{3P}{4\sqrt{2}\pi }{e}^{-\frac{{\left({\tilde{q}}_1+{\tilde{q}}_2\right)}^2}{2}}{\text{sinc}}^2\frac{P^2}{4}{\left({\tilde{q}}_1-{\tilde{q}}_2\right)}^2.$$

The covariance matrix of $p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)$ is given by

(28)$${\mathrm{\Lambda }}_{FF}^S=\left(\begin{array}{cc}{\left(\mathrm{\Delta }{\tilde{q}}_1\right)}_S^2& {〈{\tilde{q}}_1{\tilde{q}}_2〉}_S-{〈{\tilde{q}}_1〉}_S{〈{\tilde{q}}_2〉}_S\\ {〈{\tilde{q}}_2{\tilde{q}}_1〉}_S-{〈{\tilde{q}}_2〉}_S{〈{\tilde{q}}_1〉}_S& {〈\mathrm{\Delta }{\tilde{q}}_2〉}_S^2\end{array}\right).$$

The elements in the diagonal are the variances of the marginal distributions $p_{FF}^S\left({\tilde{q}}_1\right)$ and $p_{FF}^S\left({\tilde{q}}_2\right)$. In order to calculate the terms of ${\mathrm{\Lambda }}_{FF}^S$, we first note that the expected values μ_FF = {〈q̃₁〉, 〈q̃₂〉} are null. So, we have that

(29)$$\begin{array}{lll}{\left(\mathrm{\Delta }{\tilde{q}}_1\right)}_S^2={\left(\mathrm{\Delta }{\tilde{q}}_2\right)}_S^2\hfill & =\hfill & \iint d{\tilde{q}}_{1}d{\tilde{q}}_2{\tilde{q}}_1^2{p}_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\hfill \\ \hfill & =\hfill & \frac{1}{4}\left(1+\frac{3}{P^2}\right),\hfill \end{array}$$

and

(30)$$\begin{array}{lll}{〈{\tilde{q}}_1{\tilde{q}}_2〉}_S={〈{\tilde{q}}_2{\tilde{q}}_1〉}_S\hfill & =\hfill & \iint d{\tilde{q}}_{1}d{\tilde{q}}_2{\tilde{q}}_1{\tilde{q}}_2{p}_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\hfill \\ \hfill & =\hfill & \frac{1}{4}\left(1-\frac{3}{P^2}\right).\hfill \end{array}$$

Let us consider the following Gaussian approximation for describing the joint probability density distribution at the far-field plane:

(31)$$p_{FF}^G\left({\tilde{q}}_1,{\tilde{q}}_2\right)=\frac{1}{\pi {\delta }_{-}}{e}^{-\frac{{\left({\tilde{q}}_1+{\tilde{q}}_2\right)}^2}{2}}{e}^{-\frac{{\left({\tilde{q}}_1-{\tilde{q}}_2\right)}^2}{2{\delta }_{-}^2}}.$$

The elements of the covariance matrix ${\mathrm{\Lambda }}_{FF}^G$ of $p_{FF}^G\left({\tilde{q}}_1,{\tilde{q}}_2\right)$ are given by

(32)$$\begin{array}{lll}{\left(\mathrm{\Delta }{\tilde{q}}_1\right)}_G^2={\left(\mathrm{\Delta }{\tilde{q}}_2\right)}_G^2\hfill & =\hfill & \iint d{\tilde{q}}_{1}d{\tilde{q}}_2{\tilde{q}}_1^2{p}_{FF}^G\left({\tilde{q}}_1,{\tilde{q}}_2\right)\hfill \\ \hfill & =\hfill & \frac{1}{4}\left(1+{\delta }_{-}^2\right),\hfill \end{array}$$

and

(33)$$\begin{array}{lll}{〈{\tilde{q}}_1{\tilde{q}}_2〉}_G={〈{\tilde{q}}_2{\tilde{q}}_1〉}_G\hfill & =\hfill & \iint d{\tilde{q}}_{1}d{\tilde{q}}_2{\tilde{q}}_1{\tilde{q}}_2{p}_{FF}^G\left({\tilde{q}}_1,{\tilde{q}}_2\right)\hfill \\ \hfill & =\hfill & \frac{1}{4}\left(1-{\delta }_{-}^2\right).\hfill \end{array}$$

From Eq. (29), (30), (32) and (33) we can observe that the condition

(34)$${\delta }_{-}^2=\frac{3}{P^2},$$

implies that the joint distributions [Eqs. (27) and (31)] have the same covariance matrix. Hereafter, the Gaussian distribution with the same covariance matrix of Eq. (27) is denoted by $p_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)$.

It is possible to show that the differential entropy of the Gaussian distribution $p_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)$ is

(35)$$H\left[p_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]={\text{log}}_2\frac{\pi e\sqrt{3}}{P}.$$

The differential entropy of $p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)$ is given by

(36)$$\begin{array}{lll}H\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]\hfill & =\hfill & -\iint d{\tilde{q}}_{1}d{\tilde{q}}_2{p}_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right){\text{log}}_2\left(p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right)\hfill \\ \hfill & =\hfill & {\text{log}}_2\frac{4\sqrt{2}\pi }{3P}-\frac{3P}{4\sqrt{2}\pi }\iint d{\tilde{q}}_{1}d{\tilde{q}}_2\{e^{-\frac{{\left({\tilde{q}}_1+{\tilde{q}}_2\right)}^2}{2}}\hfill \\ \hfill & \hfill & \times \left[{\text{log}}_2{e}^{-\frac{{\left({\tilde{q}}_1+{\tilde{q}}_2\right)}^2}{2}}+{\text{log}}_2{\text{sinc}}^2\frac{P^2}{4}{\left({\tilde{q}}_1-{\tilde{q}}_2\right)}^2\right]\}.\hfill \end{array}$$

Making $u=\frac{{\tilde{q}}_1+{\tilde{q}}_2}{\sqrt{2}}$ and $v=\frac{P}{2}\left({\tilde{q}}_1-{\tilde{q}}_2\right)$ we obtain that

(37)$$\begin{array}{lll}H\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]\hfill & =\hfill & {\text{log}}_2\frac{4\sqrt{2}\pi }{3P}\frac{1}{2\text{ln}2}\hfill \\ \hfill & \hfill & \times \left[1-\frac{3}{\sqrt{\pi }}{\int }_0^{\infty }dv\hspace{0.17em}{\text{sinc}}^2{v}^2\text{ln}\left({\text{sinc}}^2{v}^2\right)\right].\hfill \end{array}$$

The last integral in Eq. (37) must be calculated numerically and it gives that

$${\int }_0^{\infty }dv\hspace{0.17em}{\text{sinc}}^2{v}^2\text{ln}\left({\text{sinc}}^2{v}^2\right)\approx -\mathrm{0.364.}$$

Therefore, the differential entropy of $p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)$ is

(38)$$H\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]\approx {\text{log}}_2\frac{4\sqrt{2}\pi }{3P}+\mathrm{1.17.}$$

According to Eq. (25), we have that

(39)$$\begin{array}{lll}N\left[p^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]\hfill & =\hfill & H\left[p_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]-H\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]\hfill \\ \hfill & \hfill & \approx {\text{log}}_2\frac{\pi e\sqrt{3}}{P}-{\text{log}}_2\frac{4\sqrt{2}\pi }{3P}-1.17\hfill \\ \hfill & \hfill & \approx {\text{log}}_{2}e\sqrt{\frac{27}{32}}-1.17\hfill \\ \hfill & \hfill & \approx \mathrm{0.15.}\hfill \end{array}$$

C.3. Negentropy of near-field joint distribution

The joint probability density function in the near-field plane is given by

(40)$$p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)=\frac{1}{C}{e}^{-\frac{{\left({\tilde{x}}_1+{\tilde{x}}_2\right)}^2}{2\sigma }}{\text{sint}}^2\frac{{\left({\tilde{x}}_1-{\tilde{x}}_2\right)}^2}{4P^2},$$

where C is a normalization constant given by

(41)$$C=\sqrt{2\pi }{A}_1\sqrt{\sigma }P.$$

The covariance matrix for the near-field joint distribution is

(42)$${\mathrm{\Lambda }}_{NF}^S=\left(\begin{array}{cc}{\left(\mathrm{\Delta }{\tilde{x}}_1\right)}_S^2& {〈{\tilde{x}}_1{\tilde{x}}_2〉}_S\\ {〈{\tilde{x}}_2{\tilde{x}}_1〉}_S& {\left(\mathrm{\Delta }{\tilde{x}}_2\right)}_S^2\end{array}\right),$$

where

(43)$$\begin{array}{lll}{\left(\mathrm{\Delta }{\tilde{x}}_1\right)}_S^2={\left(\mathrm{\Delta }{\tilde{x}}_2\right)}_S^2\hfill & =\hfill & \iint d{\tilde{x}}_{1}d{\tilde{x}}_2{\tilde{x}}_1^2{p}_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\hfill \\ \hfill & =\hfill & \frac{1}{4}\left(\sigma +\frac{4A_2{P}^2}{A_1}\right),\hfill \end{array}$$

and

(44)$$\begin{array}{lll}{〈{\tilde{x}}_1{\tilde{x}}_2〉}_S={〈{\tilde{x}}_2{\tilde{x}}_1〉}_S\hfill & =\hfill & \iint d{\tilde{x}}_{1}d{\tilde{x}}_2{\tilde{x}}_1{\tilde{x}}_2{p}_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\hfill \\ \hfill & =\hfill & \frac{1}{4}\left(\sigma -\frac{4A_2{P}^2}{A_1}\right).\hfill \end{array}$$

Let us now consider the following Gaussian approximation for the joint probability distribution at the near-field plane:

(45)$$p_{NF}^G\left({\tilde{x}}_1,{\tilde{x}}_2\right)=\frac{1}{\pi \sqrt{\sigma }\tau }{e}^{-\frac{{\left({\tilde{x}}_1+{\tilde{x}}_2\right)}^2}{2\sigma }}{e}^{-\frac{{\left({\tilde{x}}_1-{\tilde{x}}_2\right)}^2}{2{\tau }^2}}.$$

The covariance matrix ${\mathrm{\Lambda }}_{NF}^G$ of $p_{NF}^G\left({\tilde{x}}_1,{\tilde{x}}_2\right)$ is determined by

(46)$${\mathrm{\Lambda }}_{NF}^G=\left(\begin{array}{cc}{\left(\mathrm{\Delta }{\tilde{x}}_1\right)}_G^2& {〈{\tilde{x}}_1{\tilde{x}}_2〉}_G\\ {〈{\tilde{x}}_2{\tilde{x}}_1〉}_G& {\left(\mathrm{\Delta }{\tilde{x}}_2\right)}_G^2\end{array}\right),$$

where

(47)$$\begin{array}{lll}{\left(\mathrm{\Delta }{\tilde{x}}_1\right)}_G^2={\left(\mathrm{\Delta }{\tilde{x}}_2\right)}_G^2\hfill & =\hfill & \iint d{\tilde{x}}_{1}d{\tilde{x}}_2{\tilde{x}}_1^2{p}_{NF}^G\left({\tilde{x}}_1,{\tilde{x}}_2\right)\hfill \\ \hfill & =\hfill & \frac{1}{4}\left(\sigma +{\tau }^2\right),\hfill \end{array}$$

and

(48)$$\begin{array}{lll}{〈{\tilde{x}}_1{\tilde{x}}_2〉}_G={〈{\tilde{x}}_2{\tilde{x}}_1〉}_G\hfill & =\hfill & \iint d{\tilde{x}}_{1}d{\tilde{x}}_2{\tilde{x}}_1{\tilde{x}}_2{p}_{NF}^G\left({\tilde{x}}_1,{\tilde{x}}_2\right)\hfill \\ \hfill & =\hfill & \frac{1}{4}\left(\sigma -{\tau }^2\right).\hfill \end{array}$$

From Eq. (43), (44), (47) and (48) we can observe that the condition

(49)$${\tau }^2=\frac{4A_2{P}^2}{A_1},$$

must be satisfied for having ${\mathrm{\Lambda }}_{NF}^G={\mathrm{\Lambda }}_{NF}^S$. The Gaussian distribution which satisfies this condition is denoted by $p_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right)$.

The differential entropy of this Gaussian distribution is given by

(50)$$H\left[p_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]={\text{log}}_2\left(2\pi e\sqrt{\frac{\sigma A_2}{A_1}}P\right).$$

The differential entropy for $p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)$ reads

(51)$$\begin{array}{lll}H\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]\hfill & =\hfill & -\iint d{\tilde{x}}_{1}d{\tilde{x}}_2{p}_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right){\text{log}}_2\left(p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right)\hfill \\ \hfill & =\hfill & {\text{log}}_2\left(\sqrt{2\pi \sigma }{A}_{1}P\right)-\frac{1}{\sqrt{2\pi \sigma }{A}_{1}P}\iint d{\tilde{x}}_{1}d{\tilde{x}}_2\{e^{-\frac{{\left({\tilde{x}}_1+{\tilde{x}}_2\right)}^2}{2\sigma }}\hfill \\ \hfill & \hfill & \times {\text{sint}}^2\frac{{\left({\tilde{x}}_1-{\tilde{x}}_2\right)}^2}{4P^2}[{\text{log}}_2{e}^{-\frac{{\left({\tilde{x}}_1+{\tilde{x}}_2\right)}^2}{2\sigma }}\hfill \\ \hfill & \hfill & +{\text{log}}_2{\text{sint}}^2\frac{{\left({\tilde{x}}_1-{\tilde{x}}_2\right)}^2}{4P^2}]\}.\hfill \end{array}$$

If $r=\frac{{\tilde{x}}_1+{\tilde{x}}_2}{\sqrt{2\sigma }}$ and $s=\frac{{\tilde{x}}_1-{\tilde{x}}_2}{2P}$, we find that

(52)$$\begin{array}{lll}H\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]\hfill & =\hfill & {\text{log}}_2\left(\sqrt{2\pi \sigma }{A}_{1}P\right)\hfill \\ \hfill & \hfill & +\frac{1}{2\text{ln}2}\left[1-\frac{2}{A_1}\int ds\hspace{0.17em}\hspace{0.17em}{\text{sint}}^2{s}^2\text{ln}\hspace{0.17em}{\text{sint}}^2{s}^2\right].\hfill \end{array}$$

The integral

$$I=\int ds\hspace{0.17em}{\text{sint}}^2{s}^2\text{ln}\hspace{0.17em}{\text{sint}}^2{s}^2,$$

must be calculated numerically. It gives that I ≈ −0.692. Then, the differential entropy of the near-field joint distribution is

(53)$$H\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]\approx {\text{log}}_2\left(\sqrt{2\pi \sigma }{A}_{1}P\right)+\mathrm{1.434.}$$

Therefore, the negentropy of the near-field joint distribution is

(54)$$\begin{array}{lll}N\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]\hfill & =\hfill & H\left[p_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]-H\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]\hfill \\ \hfill & \approx \hfill & {\text{log}}_2\left(2\pi e\sqrt{\frac{\sigma A_2}{A_1}}P\right)-{\text{log}}_2\left(\sqrt{2\pi \sigma }{A}_{1}P\right)-1.434\hfill \\ \hfill & \approx \hfill & {\text{log}}_2\left(\sqrt{2\pi }e\sqrt{\frac{A_2}{A_1^3}}\right)-1.434\hfill \\ \hfill & \approx \hfill & \mathrm{0.22.}\hfill \end{array}$$

Thus, the total non-Gaussianity of SPDC is

(55)$$n{G}^T\approx 0.15+0.22=\mathrm{0.37.}$$

D. Further properties of nG^T

Property 1. nG^T = 0 iff the spatial two-photon state of SPDC is a Gaussian state.

Proof. Since

(56)$$n{G}^T=N_{NF}+N_{FF}=0,$$

and that negentropy is always nonnegative, we have that $N\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]=N\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]=0$. Thus, for the joint probabilities we have that $H\left[p_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]=H\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]$ and $H\left[p_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]=H\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]$, such that

(57)$$\begin{array}{c}\int d{\tilde{q}}_{1}d{\tilde{q}}_2{p}_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right){\text{log}}_2{p}_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)=\\ \int d{\tilde{q}}_{1}d{\tilde{q}}_2{p}_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right){\text{log}}_2{p}_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right),\end{array}$$

and

(58)$$\begin{array}{c}\int d{\tilde{x}}_{1}d{\tilde{x}}_2{p}_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right){\text{log}}_2{p}_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right)=\\ \int d{\tilde{x}}_{1}d{\tilde{x}}_2{p}_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right){\text{log}}_2{p}_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right).\end{array}$$

Since the differential entropy H is a continuous and monotonic function, it holds that

(59)$$\begin{array}{c}{p}_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right){\text{log}}_2{p}_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)=\\ p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right){\text{log}}_2{p}_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right),\end{array}$$

and

(60)$$\begin{array}{c}{p}_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right){\text{log}}_2{p}_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right)=\\ p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right){\text{log}}_2{p}_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right).\end{array}$$

Then, necessarily we have that

(61)$$p_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)=p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right),$$

and

(62)$$p_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right)=p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right).$$

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

(63)$$N\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]=N\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]=0.$$

So, the total non-Gaussianity of the spatial two-photon state becomes

(64)$$n{G}^T=0.$$

Property 2. If the spatial two-photon state is a product state then nG^T is additive, i.e., nG^T = 2 × 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(ξ₁, ξ₂) = p(ξ₁)p(ξ₂). Such type of probability density function has the covariance matrix Λ_ij defined by the elements

(65)$$\begin{array}{lll}{\mathrm{\Lambda }}_{11}\hfill & =\hfill & 〈{\xi }_1^2〉-{〈{\xi }_1〉}^2\hfill \\ \hfill & =\hfill & \int d{\xi }_1\hspace{0.17em}{\xi }_1^{2}p\left({\xi }_1\right)-{\left(\int d{\xi }_1\hspace{0.17em}{\xi }_{1}p\left({\xi }_1\right)\right)}^2,\hfill \end{array}$$

(66)$$\begin{array}{lll}{\mathrm{\Lambda }}_{22}\hfill & =\hfill & 〈{\xi }_2^2〉-{〈{\xi }_2〉}^2\hfill \\ \hfill & =\hfill & \int d{\xi }_2\hspace{0.17em}{\xi }_2^{2}p\left({\xi }_2\right)-{\left(\int d{\xi }_2\hspace{0.17em}{\xi }_{2}p\left({\xi }_2\right)\right)}^2,\hfill \end{array}$$

(67)$$\begin{array}{lll}{\mathrm{\Lambda }}_{12}\hfill & =\hfill & 〈{\xi }_1{\xi }_2〉-〈{\xi }_1〉〈{\xi }_2〉\hfill \\ \hfill & =\hfill & \int d{\xi }_1\hspace{0.17em}{\xi }_{1}p\left({\xi }_1\right)\int d{\xi }_2\hspace{0.17em}{\xi }_{2}p\left({\xi }_2\right)-\int d{\xi }_1\hspace{0.17em}{\xi }_{1}p\left({\xi }_1\right)\int d{\xi }_2\hspace{0.17em}{\xi }_{2}p\left({\xi }_2\right)\hfill \\ \hfill & =\hfill & 0,\hfill \end{array}$$

(68)$${\mathrm{\Lambda }}_{21}={\mathrm{\Lambda }}_{12}=0,$$

and a expected value vector μ = {〈ξ₁〉, 〈ξ₂〉}.

The Gaussian distribution p^G̃(ξ₁, ξ₂) with the same covariance matrix and expected value vector is given by

(69)$$\begin{array}{lll}{p}^{\tilde{G}}\left({\xi }_1,{\xi }_2\right)\hfill & =\hfill & \frac{1}{2\pi \sqrt{\text{det}\mathrm{\Lambda }}}\text{exp}\left[-\frac{1}{2}\left(\mathit{\xi }-\mathit{\mu }\right){\mathrm{\Lambda }}^{-1}{\left(\mathit{\xi }-\mathit{\mu }\right)}^T\right]\hfill \\ \hfill & =\hfill & \frac{e^{-\frac{{\left({\xi }_1-〈{\xi }_1〉\right)}^2}{2{\mathrm{\Lambda }}_{11}}}}{\sqrt{2\pi {\mathrm{\Lambda }}_{11}}}\times \frac{e^{-\frac{{\left({\xi }_2-〈{\xi }_2〉\right)}^2}{2{\mathrm{\Lambda }}_{22}}}}{\sqrt{2\pi {\mathrm{\Lambda }}_{22}}}\hfill \\ \hfill & =\hfill & p^{\tilde{G}}\left({\xi }_1\right)p^{\tilde{G}}\left({\xi }_2\right),\hfill \end{array}$$

where ξ = {ξ₁, ξ₂}.

Note that these marginal Gaussian distributions have the same variance and expected value that p(ξ₁) and p(ξ₂). Since the random variables are independent, it holds that

(70)$$H\left[p\left({\xi }_1,{\xi }_2\right)\right]=H\left[p\left({\xi }_1\right)\right]+H\left[p\left({\xi }_2\right)\right],$$

and consequently, we have that the negentropy of p(ξ₁, ξ₂) can be written as

(71)$$\begin{array}{lll}N\left[p\left({\xi }_1,{\xi }_2\right)\right]\hfill & =\hfill & H\left[p^{\tilde{G}}\left({\xi }_1,{\xi }_2\right)\right]-H\left[p\left({\xi }_1,{\xi }_2\right)\right]\hfill \\ \hfill & =\hfill & H\left[p^{\tilde{G}}\left({\xi }_1\right)\right]-H\left[p\left({\xi }_1\right)\right]+H\left[p^{\tilde{G}}\left({\xi }_2\right)\right]-H\left[p\left({\xi }_2\right)\right]\hfill \\ \hfill & =\hfill & N\left[p\left({\xi }_1\right)\right]+N\left[p\left({\xi }_2\right)\right].\hfill \end{array}$$

If the two-mode spatial state of SPDC is a product state, the near- and far-field probability density functions are written as $p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)=p_{FF}^S\left({\tilde{q}}_1\right)p_{FF}^S\left({\tilde{q}}_2\right)$ and $p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)=p_{NF}^S\left({\tilde{x}}_1\right)p_{NF}^S\left({\tilde{x}}_2\right)$. Therefore, the properties mentioned above hold. Then, the total non-Gaussianity of the spatial two-photon state of SPDC will be given by

(72)$$\begin{array}{lll}n{G}^T\hfill & =\hfill & N\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]+N\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]\hfill \\ \hfill & =\hfill & N\left[p_{FF}^S\left({\tilde{q}}_1\right)\right]+N\left[p_{FF}^S\left({\tilde{q}}_2\right)\right]+N\left[p_{NF}^S\left({\tilde{x}}_1\right)\right]+N\left[p_{NF}^S\left({\tilde{x}}_2\right)\right]\hfill \\ \hfill & =\hfill & n{G}^{M_1}+n{G}^{M_2},\hfill \end{array}$$

where we have used that $n{G}^{M_1}\equiv N\left[p_{FF}^S\left({\tilde{q}}_1\right)\right]+N\left[p_{NF}^S\left({\tilde{x}}_1\right)\right]$ and $n{G}^{M_2}\equiv N\left[p_{FF}^S\left({\tilde{q}}_2\right)\right]+N\left[p_{NF}^S\left({\tilde{x}}_2\right)\right]$.

Due to the symmetry present in the two-photon wave function [see Eq. (3) and Eq. (4) of the main paper], we have that nG^M₁ = nG^M₂ = nG^M. Then

(73)$$n{G}^T=2\times n{G}^M,$$

which proves our statement.

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

(74)$$p_{FF}^S\left({\tilde{q}}_2\right)=\int d{\tilde{q}}_1{p}_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right),$$

and

(75)$$p_{NF}^S\left({\tilde{x}}_2\right)=\int d{\tilde{x}}_1{p}_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right).$$

Doing the following change of variables: $u=\frac{P}{2}\left({\tilde{q}}_1-{\tilde{q}}_2\right)$ and $v=\frac{{\tilde{x}}_1-{\tilde{x}}_2}{2P}$, we obtain that

(76)$$p_{FF}^S\left({\tilde{q}}_2\right)\propto \int du{e}^{-\frac{2}{P^2}{\left(u+P{\tilde{q}}_2\right)}^2}{\text{sinc}}^2{u}^2,$$

and

(77)$$p_{NF}^S\left({\tilde{x}}_2\right)\propto \int dv{e}^{-2P^2{\left(v+\frac{{\tilde{x}}_2}{P}\right)}^2}{\text{sint}}^2{v}^2.$$

Considering the limiting case (i) we may write the marginal distributions as

(78)$$p_{FF}^S\left({\tilde{q}}_2\right)\propto \int du\delta \left(u+P{\tilde{q}}_2\right){\text{sinc}}^2{u}^2\propto {\text{sinc}}^2\left(P^2{\tilde{q}}_2\right),$$

and

(79)$$p_{NF}^S\left({\tilde{x}}_2\right)\propto e^{-2{\tilde{x}}_2^2}.$$

In such case, it is clear that the negentropy of $p_{NF}^S\left({\tilde{x}}_2\right)$ is null $\left[N\left[p_{NF}^S\left({\tilde{x}}_2\right)\right]=0\right]$. The negentropy of the far-field marginal distribution will be the same of the function sinc²(P²q̃₂). It gives that

(80)$$N\left[p_{FF}^S\left({\tilde{q}}_2\right)\right]\approx {\text{log}}_2\left(\sqrt{\frac{27e}{32}}\right)-0.4447=\mathrm{0.154.}$$

Then, for values of P ≪ 1, nG^M ≈ 0.154 such that nG^T > nG^M.

In the other limiting case (ii), we have that

(81)$$p_{FF}^S\left({\tilde{q}}_2\right)\propto e^{-2{\tilde{q}}_2^2},$$

and

(82)$$p_{NF}^S\left({\tilde{x}}_2\right)\propto \int dv\delta \left(v+\frac{{\tilde{x}}_2}{P}\right){\text{sint}}^2{v}^2\propto {\text{sint}}^2\frac{{\tilde{x}}_2^2}{P^2}.$$

Now, the negentropy of $p_{FF}^S\left({\tilde{q}}_2\right)$ is null $\left[N\left[p_{FF}^S\left({\tilde{q}}_2\right)\right]=0\right]$. The negentropy of the near-field marginal distribution will be the same of the function ${\text{sint}}^2\frac{{\tilde{x}}_2^2}{P^2}$. It gives that

(83)$$N\left[p_{NF}^S\left({\tilde{x}}_2\right)\right]\approx {\text{log}}_2\left(\sqrt{\frac{2\pi e{A}_2}{A_1^3}}\right)-0.7122=0.224$$

Thus, for larger values of P, nG^M ≈ 0.224 which is also smaller than nG^T.

Intermediate cases, for values of 0.002 < 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^M is plotted in terms of P. As one can see, in such intermediate cases, one always have that nG^T > nG^M. For P = 3 the value of nG^M ≈ 0.175. For values of P > 9, nG^M already tends to its maximal value which is $n{G}_{\mathit{max}}^M=0.224$ (See Fig. 5 below). This proves that the non-Gaussianity of the spatial state of SPDC decreases under partial trace.

 

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

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E. Calculating the non-Gaussianity of the state of the spatially correlated down-converted photons using the QRE

For comparison purposes, we calculated the value of the non-Gaussianity measure based on the QRE, δ_B [27, 28]. This was done by assuming the case of perfect phase-matching to simplify the calculation of the mean values in the covariance matrix. Then, we considered that no correlations are present between the position of one photon and the momentum of the other photon. Therefore, the cross joint probabilities p(x_i, q_j) (where i, j = 1, 2 and i ≠ j) are described by a product of the marginal position and momentum distributions p(x_i) and p(q_j). In such case the covariance matrix can be written, considering the order V(x̃₁, q̃₁, x̃₂, q̃₂), as

(84)$$\mathbf{V}=\left(\begin{array}{cccc}\frac{1}{4}\left(1+\frac{4A_2{P}^2}{A_1}\right)& 0& \frac{1}{4}\left(1-\frac{4A_2{P}^2}{A_1}\right)& 0\\ 0& \frac{1}{4}\left(1+\frac{3}{P^2}\right)& 0& \frac{1}{4}\left(1-\frac{3}{P^2}\right)\\ \frac{1}{4}\left(1-\frac{4A_2{P}^2}{A_1}\right)& 0& \frac{1}{4}\left(1+\frac{4A_2{P}^2}{A_1}\right)& 0\\ 0& \frac{1}{4}\left(1-\frac{3}{P^2}\right)& 0& \frac{1}{4}\left(1+\frac{3}{P^2}\right)\end{array}\right).$$

The Gaussian state associated to this covariance matrix has a purity of μ = 0.44 [28], and its Von-Neumann entropy reads S = 1.08. Since the state of Eq. (1) is pure, we have that δ_B = 1.08, which does not depend on the value of P.

F. nG^T at the limit when P ≪ 1

Here we discuss the behavior of nG^T when considering the limit of 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:

(85)$$N\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]=H\left[p_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]-H\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right].$$

In the limits considered, the differential entropies can be written as H[p(ξ₁, ξ₂)] = H[p(ξ₁|ξ₂)] + H[p(ξ₂)], such that the negentropy of the far-field joint distribution may also be rewritten as

(86)$$\begin{array}{lll}N\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]\hfill & =\hfill & H\left[p_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]+H\left[p_{FF}^{\tilde{G}}\left({\tilde{q}}_2\right)\right]\hfill \\ \hfill & \hfill & -H\left[p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2\right)\right]-H\left[p_{FF}^S\left({\tilde{q}}_2\right)\right]\hfill \\ \hfill & =\hfill & H\left[p_{FF}^{\tilde{G}}\left({\tilde{q}}_1|{\tilde{q}}_2\right)\right]-H\left[p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2\right)\right]\hfill \\ \hfill & \hfill & +N\left[p_{FF}^S\left({\tilde{q}}_2\right)\right].\hfill \end{array}$$

Note that $p_{FF}^{\tilde{G}}\left({\tilde{q}}_1|{\tilde{q}}_2\right)$ does not have the same expected value and variance of the conditional probability density function $p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2\right)$. Thus, in general, we can not define $H\left[p^{\tilde{G}}\left({\tilde{q}}_1|{\tilde{q}}_2\right)\right]-H\left[p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2\right)\right]$ as the negentropy of the conditional far-field distribution.

Considering the joint far-field distribution Eq. (27), the conditional distribution when q̃₂ = 0 will be given by

(87)$$p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2=0\right)\propto e^{-\frac{{\tilde{q}}_1^2}{2}}{\text{sinc}}^2\frac{P^2{\tilde{q}}_1^2}{4}.$$

When P ≪ 1, the sinc function is much larger than the Gaussian function, such that we can approximate this distribution as

(88)$$p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2=0\right)\approx \frac{1}{\sqrt{2\pi }}{e}^{-\frac{{\tilde{q}}_1^2}{2}}.$$

Therefore, the differential entropy is

(89)$$H\left[p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2=0\right)\right]\approx \frac{1}{2}{\text{log}}_2\left(2\pi e\right).$$

Note that Eq. (88) is a normal distribution with the expected value equal to zero and the variance equal to 1.

Now let’s consider the Gaussian distribution $p_{FF}^{\tilde{G}}\left({\tilde{q}}_1,{\tilde{q}}_2\right)$ of Eq. (31). It gives the conditional Gaussian distribution $p_{FF}^{\tilde{G}}\left({\tilde{q}}_1|{\tilde{q}}_2\right)$ at q̃₂ = 0 equal to

(90)$$p_{FF}^{\tilde{G}}\left({\tilde{q}}_1|{\tilde{q}}_2=0\right)=\frac{\sqrt{3+P^2}}{\sqrt{6\pi }}{e}^{-{\tilde{q}}_1^2\frac{3+P^2}{6}}.$$

In the case when 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 $p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2=0\right)$. Hence, in this limit, we can define the negentropy of the far-field conditional distribution in terms of the $p_{FF}^{\tilde{G}}\left({\tilde{q}}_1|{\tilde{q}}_2=0\right)$, such that

(91)$$N\left[p_{FF}^S\left({\tilde{q}}_1,{\tilde{q}}_2\right)\right]=N\left[p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2\right)\right]+N\left[p_{FF}^S\left({\tilde{q}}_2\right)\right].$$

Now we consider the negentropy of the near-field joint distribution Eq. (40). If P ≪ 1, then the near-field conditional distribution is given by

(92)$$p_{NF}^S\left({\tilde{x}}_1|{\tilde{x}}_2=0\right)\approx \frac{1}{2P{A}_1}{\text{sint}}^2\frac{{\tilde{x}}_1^2}{4P^2},$$

where we assumed σ = 1 for simplicity.

The variance of this function is

(93)$${\left({\mathrm{\Delta }{\tilde{x}}_1|}_{{\tilde{x}}_2=0}\right)}^2=\int d{\tilde{x}}_1{\tilde{x}}_1^2{p}_{NF}^S\left({\tilde{x}}_1|{\tilde{x}}_2=0\right)=\frac{4A_2{P}^2}{A_1},$$

where A₁ and A₂ are the constants defined above. Now we consider the near-field Gaussian joint distribution $p_{NF}^{\tilde{G}}\left({\tilde{x}}_1,{\tilde{x}}_2\right)$. The near-field conditional distribution associated with this Gaussian approximation, when P ≪ 1, is given by

(94)$$p_{NF}^{\tilde{G}}\left({\tilde{x}}_1|{\tilde{x}}_2=0\right)=\frac{\sqrt{A_1}}{2\sqrt{2\pi A_2}P}{e}^{-\frac{A_1}{8A_2{P}^2}{\tilde{x}}_1^2}.$$

As one can note, $p_{NF}^{\tilde{G}}\left({\tilde{x}}_1|{\tilde{x}}_2=0\right)$ has the same expected value and variance of $p_{NF}^S\left({\tilde{x}}_1|{\tilde{x}}_2=0\right)$. Therefore, in this limit we can write the negentropy of the near-field joint distribution as

(95)$$N\left[p_{NF}^S\left({\tilde{x}}_1,{\tilde{x}}_2\right)\right]=N\left[p_{NF}^S\left({\tilde{x}}_1|{\tilde{x}}_2\right)\right]+N\left[p_{NF}^S\left({\tilde{x}}_2\right)\right],$$

where $N\left[p_{NF}^S\left({\tilde{x}}_1|{\tilde{x}}_2\right)\right]$ is defined in terms of the differential entropy of $p_{NF}^{\tilde{G}}\left({\tilde{x}}_1|{\tilde{x}}_2\right)$.

Thus, when we consider P ≪ 1 we can write the total non-Gaussianity of the spatial two-photon state of SPDC as

(96)$$n{G}^T\approx n{G}^C+n{G}^M\left(\text{iff}\hspace{0.17em}P\ll 1\right),$$

where $n{G}^C=N\left[p_{NF}^S\left({\tilde{x}}_1|{\tilde{x}}_2\right)\right]+N\left[p_{FF}^S\left({\tilde{q}}_1|{\tilde{q}}_2\right)\right]$, and $n{G}^M=N\left[p_{FF}^S\left({\tilde{q}}_j\right)\right]+N\left[p_{NF}^S\left({\tilde{x}}_j\right)\right]$, with j = 1,2.

Acknowledgments

We thank M. Martinelli, J. L. Romero, C. Saavedra and L. Neves for discussions related with earlier drafts of this paper. We acknowledge the support of Grants FONDECYT 1120067, Milenio P10-030-F and PFB 08024. C. H. M. acknowledges CNPq and CAPES. E. S. G. acknowledges the financial support of CONICYT.

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