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

doi:10.1364/OE.20.003753

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

E. S. Gómez, W. A. T. Nogueira, C. H. Monken, and G. Lima »View Author Affiliations

Optics Express, Vol. 20, Issue 4, pp. 3753-3772 (2012)
http://dx.doi.org/10.1364/OE.20.003753

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▸ Article Outline
– Abstract
1. Introduction
2. The phase-matching conditions and the variances of the conditional probabilities
3. The EPR-Criterion as a witness for the non-Gaussianity of the spatial two-photon state
4. Quantifying the non-Gaussianity of the spatial two-photon state of SPDC
5. Conclusion
– References and links

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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.

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

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

, 2

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

]. This type of Einstein-Podolsky-Rosen (EPR) correlation [3

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

], has been used as a resource for fundamental studies of quantum mechanics [1

, 2

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

, 4

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

–6

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

], for quantum imaging [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 52, R3429 (1995). [CrossRef] [PubMed]

, 8

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

], and experiments of quantum information [9

L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005). [CrossRef] [PubMed]

–11

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

The process of SPDC has been studied extensively in the past [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]

], 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

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]

]. 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

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]

]. 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

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]

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

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]

] 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

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]

], 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

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

] 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

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]

], 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

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]

]. 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

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]

]. The quantification of the non-Gaussianity of a quantum state has important applications for quantum information [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]

], 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

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]

]. In the momentum representation, the two-photon state is given by [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]

]

| Ψ 〉 \propto \iint d q_{1} d q_{2} e^{- \frac{ζ}{4} {| q_{1} + q_{2} |}^{2}} sinc \frac{L {| q_{1} - q_{2} |}^{2}}{4 k_{p}} | q_{1}, q_{2} 〉,

(1)

where |q₁, q₂〉 represents a two-photon state in plane-wave modes whose transverse wave vectors are q₁ and q₂. L is the crystal length, k_p is the pump beam wave number,

ζ = w_{0}^{2} - 2 i z / k_{p}

, and w₀ is the pump beam waist, which is located at z = 0. This state may be rewritten in the coordinate space as [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]

]

| Ψ 〉 \propto \iint d ρ_{1} d ρ_{2} e^{- \frac{1}{4 ζ} {| ρ_{1} + ρ_{2} |}^{2}} sint \frac{k_{p} {| ρ_{1} - ρ_{2} |}^{2}}{4 L} | ρ_{1}, ρ_{2} 〉,

(2)

where we define the function sint as

sint (x) \equiv \frac{2}{π} \int_{x}^{\infty} d t sinc t \equiv 1 - \frac{2}{π} Si (x)

, Si(x) being the sine integral function. The functions sinc(b|q|²/2) and

\frac{π}{2 b} sint ({| ρ |}^{2} / 2 b)

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(bx²) by e^−αbx². Sometimes it is used that α = 1 [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]

], and in other cases the value of α is chosen such that both functions coincide at 1/e² [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]

] or at 1/e [22

] of their peak. While this approximation seems to be reasonable for the entanglement quantification [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]

], 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

p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) \propto e^{- \frac{1}{2} {({\tilde{q}}_{1} + {\tilde{q}}_{2})}^{2}} {(sinc \frac{P^{2} {({\tilde{q}}_{1} - {\tilde{q}}_{2})}^{2}}{4})}^{2},

(3)

p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) \propto e^{- \frac{1}{2 σ^{2}} {({\tilde{x}}_{1} + {\tilde{x}}_{2})}^{2}} {(sint \frac{{({\tilde{x}}_{1} - {\tilde{x}}_{2})}^{2}}{4 P^{2}})}^{2},

(4)

where

σ^{2} = (z_{0}^{2} + z^{2}) / z_{0}^{2}

P = \sqrt{L / (2 z_{0})}

, and

z_{0} = k_{p} w_{0}^{2} / 2

is the diffraction length of the pump beam. The joint probabilities are related with the conditional (and marginal) probabilities

p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2}) [p_{F F}^{S} ({\tilde{q}}_{j})]

and

p_{N F}^{S} ({\tilde{x}}_{1} | {\tilde{x}}_{2}) [p_{N F}^{S} ({\tilde{x}}_{j})]

, through the rule:

p (ξ_{1} | ξ_{2}) = \frac{p (ξ_{1}, ξ_{2})}{p (ξ_{2})}

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

p_{F F, α_{i}}^{G} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) \propto e^{- \frac{1}{2} {({\tilde{q}}_{1} + {\tilde{q}}_{2})}^{2}} e^{- α_{i} \frac{P^{2}}{2} {({\tilde{q}}_{1} - {\tilde{q}}_{2})}^{2}},

(5)

p_{N F, α_{i}}^{G} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) \propto e^{- \frac{1}{2 σ^{2}} {({\tilde{x}}_{1} + {\tilde{x}}_{2})}^{2}} e^{- \frac{1}{2 α_{i} P^{2}} {({\tilde{x}}_{1} - {\tilde{x}}_{2})}^{2}},

(6)

where different values of α_i represent distinct Gaussian approximations for the sinc function. In Fig. 1(a) [(b)] we compare the curves

p_{F F}^{S}

and

p_{F F, α_{i}}^{G}

(

p_{N F}^{S}

and

p_{N F, α_{i}}^{G}

) considering x̃₂, q̃₂ = 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

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]

,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]

], 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 w₀. In this case we find α₁ = 0.45 (α₂ = 0.72) for the case where the sinc and Gaussian functions coincide at 1/e (1/e²). 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- [(Δq₁|_q₂)²L/k_p] and near-field [(Δx₁|_x₂)²k_p/L] conditional distributions are plotted in terms of P.

Fig. 1 The far- and near-field conditional probability distributions, while considering the state of SPDC (

p_{F F}^{S}

and

p_{N F}^{S}

) and when some Gaussian approximations are assumed (

p_{F F, α_{i}}^{G}

and

p_{N F, α_{i}}^{G}

). In (a)–(d) the curves

p_{F F}^{S}

and

p_{N F}^{S}

(red solid lines) are compared with

p_{F F, α_{2}}^{G}

and

p_{N F, α_{2}}^{G}

for α₁ = 0.45 (green dashed line), α₂ = 0.72 (blue dot-dashed line), and for some specific values of P. In (e) and (f) the variances of the momentum and position conditional probabilities are plotted in terms of P, respectively. α₃ = 1.

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

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

]. This is done by observing the violation of the inequality

{({Δ x_{1} |}_{x_{2}})}^{2} {({Δ q_{1} |}_{q_{2}})}^{2} \geq \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.

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

{({Δ x_{1} |}_{x_{2}})}^{2} {({Δ q_{1} |}_{q_{2}})}^{2} = \frac{1}{4 K}

[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

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]

] 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

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

], 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

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

]. 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^G^̃(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

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

]. 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

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

Motivated by these properties we define the total non-Gaussianity of the spatial two-photon state of SPDC as

n G^{T} \equiv N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] + N [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})],

(7)

where

N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})]

and

N [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})]

are the negentropies of the far- and near-field joint distributions of Eq. (3) and Eq. (4), respectively. In Appendix C we give, explicitly, the calculations of

N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})]

and

N [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})]

. It is possible to observe that nG^T = 0 if and only if |Ψ〉 is a two-mode Gaussian state [see Appendix D]. Otherwise nG^T > 0. We obtain that

N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] \approx 0.15

and that

N [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] \approx 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

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].

]. The fact that nG^T ≠ 0, emphasizes that spatial Gaussian approximations should be taken carefully due to the extremality of Gaussian states [35

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

]. 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

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]

]. 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:

n G^{C} \equiv N [p_{F F}^{S} (q_{i} | q_{j})] + N [p_{N F}^{S} (x_{i} | x_{j})]

and

n G^{M} \equiv N [p_{F F}^{S} (q_{i})] + N [p_{N F}^{S} (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

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]

In Fig. 3(a) we plot the negentropies

N [p_{F F}^{S} (q_{1} | q_{2})]

and

N [p_{N F}^{S} (x_{1} | x_{2})]

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.

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:

n G^{T} \approx n G^{C} + n G^{M} (iff P ≪ 1) .

(8)

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.

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

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

] 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

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]

], 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.

Appendices

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

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]

]

| Ψ 〉 = \int d q_{1} \int d q_{2} Ψ (q_{1}, q_{2}) | 1 q_{1} 〉 | 1 q_{2} 〉,

(9)

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

Ψ (q_{1}, q_{2}) \propto Δ (q_{1} + q_{2}) Θ (q_{1} - q_{2}),

(10)

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

Δ (q_{1} + q_{2}) = exp [- \frac{{(q_{1} + q_{2})}^{2}}{2 σ_{+}^{2}}]

and

Θ (q_{1} - q_{2}) = exp [- \frac{{(q_{1} - q_{2})}^{2}}{2 δ_{-}^{2}}]

[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]

, 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

], we have in momentum representation that

Ψ (q_{1}, q_{2}) \propto e^{- {(q_{1} + q_{2})}^{2} / 2 σ_{+}^{2}} e^{- {(q_{1} - q_{2})}^{2} / 2 δ_{-}^{2}},

(11)

and in position representation that

Ψ (x_{1}, x_{2}) \propto e^{- σ_{+}^{2} {(x_{1} + x_{2})}^{2} / 8} e^{- δ_{-}^{2} {(x_{1} - x_{2})}^{2} / 8} .

(12)

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 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

P (x_{1} | x_{2} = 0) = \frac{\sqrt{σ_{+}^{2} + δ_{-}^{2}}}{2 \sqrt{π}} exp [- \frac{σ_{+}^{2} + δ_{-}^{2}}{4} x_{1}^{2}],

(13)

and

P (q_{1} | q_{2} = 0) = \frac{\sqrt{σ_{+}^{2} + δ_{-}^{2}}}{\sqrt{π} σ_{+}^{2} δ_{-}^{2}} exp [- \frac{σ_{+}^{2} + δ_{-}^{2}}{σ_{+}^{2} δ_{-}^{2}} q_{1}^{2}] .

(14)

Here

σ_{+}^{2}

and

δ_{-}^{2}

are the widths of the respective Gaussian functions [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]

]. The variances of the conditional distributions can be calculated directly from Eqs. (13) and (14), which results in

{({Δ x_{1} |}_{x_{2}})}^{2} = \frac{2}{σ_{+}^{2} + δ_{-}^{2}},

(15)

and

{({Δ q_{1} |}_{q_{2}})}^{2} = \frac{σ_{+}^{2} δ_{-}^{2}}{2 (σ_{+}^{2} + δ_{-}^{2})} .

(16)

Then, the product of them is given by

{({Δ x_{1} |}_{x_{2}})}^{2} {({Δ q_{1} |}_{q_{2}})}^{2} = \frac{σ_{+}^{2} δ_{-}^{2}}{{(σ_{+}^{2} + δ_{-}^{2})}^{2}} .

(17)

Let us now consider the Schmidt number K for Gaussian states. It was showed in Ref. [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]

] that it is given by the expression

K = \frac{1}{4} {(\frac{σ_{+}}{δ_{-}} + \frac{δ_{-}}{σ_{+}})}^{2} = \frac{1}{4} \frac{{(σ_{+}^{2} + δ_{-}^{2})}^{2}}{σ_{+}^{2} δ_{-}^{2}} .

(18)

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

{({Δ x_{1} |}_{x_{2}})}^{2} {({Δ q_{1} |}_{q_{2}})}^{2} = \frac{1}{4 K} .

(19)

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

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

]. For the case of spatial entanglement it reads [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]

]

{[Δ ({\tilde{x}}_{1} - {\tilde{x}}_{2})]}^{2} {[Δ ({\tilde{q}}_{1} + {\tilde{q}}_{2})]}^{2} \geq 1.

(20)

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

{[Δ ({\tilde{q}}_{1} + {\tilde{q}}_{2})]}^{2} \propto \int d {\tilde{q}}_{+} {\tilde{q}}_{+}^{2} e^{- \frac{1}{2} {\tilde{q}}_{+}^{2}} = 1,

(21)

{[Δ ({\tilde{x}}_{1} - {\tilde{x}}_{2})]}^{2} \propto \int d {\tilde{x}}_{-} {\tilde{x}}_{-}^{2} {sint}^{2} \frac{1}{4 P} {\tilde{x}}_{-}^{2} = 4 \frac{A_{2}}{A_{1}} P^{2},

(22)

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

A_{1} = \int_{- \infty}^{\infty} d ξ {sint}^{2} ξ^{2} \approx 1.4008,

(23)

and

A_{2} = \int_{- \infty}^{\infty} d ξ ξ^{2} {sint}^{2} ξ^{2} \approx 0.5897.

(24)

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].

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

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

]

N \equiv H [p^{\tilde{G}} (ξ_{1}, ξ_{2})] - H [p (ξ_{1}, ξ_{2})],

(25)

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

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

]

H [p (ξ_{1}, ξ_{2})] \equiv - \int d ξ_{1} \int d ξ_{2} p (ξ_{1}, ξ_{2}) {log}_{2} (p (ξ_{1}, ξ_{2})) .

(26)

C.2. Negentropy of far-field joint distribution

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

p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) = \frac{3 P}{4 \sqrt{2} π} e^{- \frac{{({\tilde{q}}_{1} + {\tilde{q}}_{2})}^{2}}{2}} {sinc}^{2} \frac{P^{2}}{4} {({\tilde{q}}_{1} - {\tilde{q}}_{2})}^{2} .

(27)

The covariance matrix of

p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})

is given by

Λ_{F F}^{S} = (\begin{matrix} {(Δ {\tilde{q}}_{1})}_{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} & {〈 Δ {\tilde{q}}_{2} 〉}_{S}^{2} \end{matrix}) .

(28)

The elements in the diagonal are the variances of the marginal distributions

p_{F F}^{S} ({\tilde{q}}_{1})

and

p_{F F}^{S} ({\tilde{q}}_{2})

. In order to calculate the terms of

Λ_{F F}^{S}

, we first note that the expected values μ_FF = {〈q̃₁〉, 〈q̃₂〉} are null. So, we have that

\begin{array}{l} {(Δ {\tilde{q}}_{1})}_{S}^{2} = {(Δ {\tilde{q}}_{2})}_{S}^{2} & = & \iint d {\tilde{q}}_{1} d {\tilde{q}}_{2} {\tilde{q}}_{1}^{2} p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) \\ = & \frac{1}{4} (1 + \frac{3}{P^{2}}), \end{array}

(29)

and

\begin{array}{l} {〈 {\tilde{q}}_{1} {\tilde{q}}_{2} 〉}_{S} = {〈 {\tilde{q}}_{2} {\tilde{q}}_{1} 〉}_{S} & = & \iint d {\tilde{q}}_{1} d {\tilde{q}}_{2} {\tilde{q}}_{1} {\tilde{q}}_{2} p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) \\ = & \frac{1}{4} (1 - \frac{3}{P^{2}}) . \end{array}

(30)

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

p_{F F}^{G} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) = \frac{1}{π δ_{-}} e^{- \frac{{({\tilde{q}}_{1} + {\tilde{q}}_{2})}^{2}}{2}} e^{- \frac{{({\tilde{q}}_{1} - {\tilde{q}}_{2})}^{2}}{2 δ_{-}^{2}}} .

(31)

The elements of the covariance matrix

Λ_{F F}^{G}

p_{F F}^{G} ({\tilde{q}}_{1}, {\tilde{q}}_{2})

are given by

\begin{array}{l} {(Δ {\tilde{q}}_{1})}_{G}^{2} = {(Δ {\tilde{q}}_{2})}_{G}^{2} & = & \iint d {\tilde{q}}_{1} d {\tilde{q}}_{2} {\tilde{q}}_{1}^{2} p_{F F}^{G} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) \\ = & \frac{1}{4} (1 + δ_{-}^{2}), \end{array}

(32)

and

\begin{array}{l} {〈 {\tilde{q}}_{1} {\tilde{q}}_{2} 〉}_{G} = {〈 {\tilde{q}}_{2} {\tilde{q}}_{1} 〉}_{G} & = & \iint d {\tilde{q}}_{1} d {\tilde{q}}_{2} {\tilde{q}}_{1} {\tilde{q}}_{2} p_{F F}^{G} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) \\ = & \frac{1}{4} (1 - δ_{-}^{2}) . \end{array}

(33)

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

δ_{-}^{2} = \frac{3}{P^{2}},

(34)

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_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2})

It is possible to show that the differential entropy of the Gaussian distribution

p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2})

H [p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] = {log}_{2} \frac{π e \sqrt{3}}{P} .

(35)

The differential entropy of

p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})

is given by

\begin{array}{l} H [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] & = & - \iint d {\tilde{q}}_{1} d {\tilde{q}}_{2} p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) {log}_{2} (p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})) \\ = & {log}_{2} \frac{4 \sqrt{2} π}{3 P} - \frac{3 P}{4 \sqrt{2} π} \iint d {\tilde{q}}_{1} d {\tilde{q}}_{2} {e^{- \frac{{({\tilde{q}}_{1} + {\tilde{q}}_{2})}^{2}}{2}} \\ \times [{log}_{2} e^{- \frac{{({\tilde{q}}_{1} + {\tilde{q}}_{2})}^{2}}{2}} + {log}_{2} {sinc}^{2} \frac{P^{2}}{4} {({\tilde{q}}_{1} - {\tilde{q}}_{2})}^{2}]} . \end{array}

(36)

Making

u = \frac{{\tilde{q}}_{1} + {\tilde{q}}_{2}}{\sqrt{2}}

and

v = \frac{P}{2} ({\tilde{q}}_{1} - {\tilde{q}}_{2})

we obtain that

\begin{array}{l} H [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] & = & {log}_{2} \frac{4 \sqrt{2} π}{3 P} \frac{1}{2 ln 2} \\ \times [1 - \frac{3}{\sqrt{π}} \int_{0}^{\infty} d v {sinc}^{2} v^{2} ln ({sinc}^{2} v^{2})] . \end{array}

(37)

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

\int_{0}^{\infty} d v {sinc}^{2} v^{2} ln ({sinc}^{2} v^{2}) \approx - 0.364.

Therefore, the differential entropy of

p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})

H [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] \approx {log}_{2} \frac{4 \sqrt{2} π}{3 P} + 1.17.

(38)

According to Eq. (25), we have that

\begin{array}{l} N [p^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] & = & H [p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] - H [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] \\ \approx {log}_{2} \frac{π e \sqrt{3}}{P} - {log}_{2} \frac{4 \sqrt{2} π}{3 P} - 1.17 \\ \approx {log}_{2} e \sqrt{\frac{27}{32}} - 1.17 \\ \approx 0.15. \end{array}

(39)

C.3. Negentropy of near-field joint distribution

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

p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) = \frac{1}{C} e^{- \frac{{({\tilde{x}}_{1} + {\tilde{x}}_{2})}^{2}}{2 σ}} {sint}^{2} \frac{{({\tilde{x}}_{1} - {\tilde{x}}_{2})}^{2}}{4 P^{2}},

(40)

where C is a normalization constant given by

C = \sqrt{2 π} A_{1} \sqrt{σ} P .

(41)

The covariance matrix for the near-field joint distribution is

Λ_{N F}^{S} = (\begin{matrix} {(Δ {\tilde{x}}_{1})}_{S}^{2} & {〈 {\tilde{x}}_{1} {\tilde{x}}_{2} 〉}_{S} \\ {〈 {\tilde{x}}_{2} {\tilde{x}}_{1} 〉}_{S} & {(Δ {\tilde{x}}_{2})}_{S}^{2} \end{matrix}),

(42)

where

\begin{array}{l} {(Δ {\tilde{x}}_{1})}_{S}^{2} = {(Δ {\tilde{x}}_{2})}_{S}^{2} & = & \iint d {\tilde{x}}_{1} d {\tilde{x}}_{2} {\tilde{x}}_{1}^{2} p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) \\ = & \frac{1}{4} (σ + \frac{4 A_{2} P^{2}}{A_{1}}), \end{array}

(43)

and

\begin{array}{l} {〈 {\tilde{x}}_{1} {\tilde{x}}_{2} 〉}_{S} = {〈 {\tilde{x}}_{2} {\tilde{x}}_{1} 〉}_{S} & = & \iint d {\tilde{x}}_{1} d {\tilde{x}}_{2} {\tilde{x}}_{1} {\tilde{x}}_{2} p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) \\ = & \frac{1}{4} (σ - \frac{4 A_{2} P^{2}}{A_{1}}) . \end{array}

(44)

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

p_{N F}^{G} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) = \frac{1}{π \sqrt{σ} τ} e^{- \frac{{({\tilde{x}}_{1} + {\tilde{x}}_{2})}^{2}}{2 σ}} e^{- \frac{{({\tilde{x}}_{1} - {\tilde{x}}_{2})}^{2}}{2 τ^{2}}} .

(45)

The covariance matrix

Λ_{N F}^{G}

p_{N F}^{G} ({\tilde{x}}_{1}, {\tilde{x}}_{2})

is determined by

Λ_{N F}^{G} = (\begin{matrix} {(Δ {\tilde{x}}_{1})}_{G}^{2} & {〈 {\tilde{x}}_{1} {\tilde{x}}_{2} 〉}_{G} \\ {〈 {\tilde{x}}_{2} {\tilde{x}}_{1} 〉}_{G} & {(Δ {\tilde{x}}_{2})}_{G}^{2} \end{matrix}),

(46)

where

\begin{array}{l} {(Δ {\tilde{x}}_{1})}_{G}^{2} = {(Δ {\tilde{x}}_{2})}_{G}^{2} & = & \iint d {\tilde{x}}_{1} d {\tilde{x}}_{2} {\tilde{x}}_{1}^{2} p_{N F}^{G} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) \\ = & \frac{1}{4} (σ + τ^{2}), \end{array}

(47)

and

\begin{array}{l} {〈 {\tilde{x}}_{1} {\tilde{x}}_{2} 〉}_{G} = {〈 {\tilde{x}}_{2} {\tilde{x}}_{1} 〉}_{G} & = & \iint d {\tilde{x}}_{1} d {\tilde{x}}_{2} {\tilde{x}}_{1} {\tilde{x}}_{2} p_{N F}^{G} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) \\ = & \frac{1}{4} (σ - τ^{2}) . \end{array}

(48)

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

τ^{2} = \frac{4 A_{2} P^{2}}{A_{1}},

(49)

must be satisfied for having

Λ_{N F}^{G} = Λ_{N F}^{S}

. The Gaussian distribution which satisfies this condition is denoted by

p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2})

The differential entropy of this Gaussian distribution is given by

H [p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] = {log}_{2} (2 π e \sqrt{\frac{σ A_{2}}{A_{1}}} P) .

(50)

The differential entropy for

p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})

reads

\begin{array}{l} H [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] & = & - \iint d {\tilde{x}}_{1} d {\tilde{x}}_{2} p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) {log}_{2} (p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})) \\ = & {log}_{2} (\sqrt{2 π σ} A_{1} P) - \frac{1}{\sqrt{2 π σ} A_{1} P} \iint d {\tilde{x}}_{1} d {\tilde{x}}_{2} {e^{- \frac{{({\tilde{x}}_{1} + {\tilde{x}}_{2})}^{2}}{2 σ}} \\ \times {sint}^{2} \frac{{({\tilde{x}}_{1} - {\tilde{x}}_{2})}^{2}}{4 P^{2}} [{log}_{2} e^{- \frac{{({\tilde{x}}_{1} + {\tilde{x}}_{2})}^{2}}{2 σ}} \\ + {log}_{2} {sint}^{2} \frac{{({\tilde{x}}_{1} - {\tilde{x}}_{2})}^{2}}{4 P^{2}}]} . \end{array}

(51)

r = \frac{{\tilde{x}}_{1} + {\tilde{x}}_{2}}{\sqrt{2 σ}}

and

s = \frac{{\tilde{x}}_{1} - {\tilde{x}}_{2}}{2 P}

, we find that

\begin{array}{l} H [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] & = & {log}_{2} (\sqrt{2 π σ} A_{1} P) \\ + \frac{1}{2 ln 2} [1 - \frac{2}{A_{1}} \int d s {sint}^{2} s^{2} ln {sint}^{2} s^{2}] . \end{array}

(52)

The integral

I = \int d s {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

H [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] \approx {log}_{2} (\sqrt{2 π σ} A_{1} P) + 1.434.

(53)

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

\begin{array}{l} N [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] & = & H [p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] - H [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] \\ \approx & {log}_{2} (2 π e \sqrt{\frac{σ A_{2}}{A_{1}}} P) - {log}_{2} (\sqrt{2 π σ} A_{1} P) - 1.434 \\ \approx & {log}_{2} (\sqrt{2 π} e \sqrt{\frac{A_{2}}{A_{1}^{3}}}) - 1.434 \\ \approx & 0.22. \end{array}

(54)

Thus, the total non-Gaussianity of SPDC is

n G^{T} \approx 0.15 + 0.22 = 0.37.

(55)

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

n G^{T} = N_{N F} + N_{F F} = 0,

(56)

and that negentropy is always nonnegative, we have that

N [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] = N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] = 0

. Thus, for the joint probabilities we have that

H [p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] = H [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})]

and

H [p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] = H [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})]

, such that

\begin{matrix} \int d {\tilde{q}}_{1} d {\tilde{q}}_{2} p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) {log}_{2} p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) = \\ \int d {\tilde{q}}_{1} d {\tilde{q}}_{2} p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) {log}_{2} p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}), \end{matrix}

(57)

and

\begin{matrix} \int d {\tilde{x}}_{1} d {\tilde{x}}_{2} p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) {log}_{2} p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) = \\ \int d {\tilde{x}}_{1} d {\tilde{x}}_{2} p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) {log}_{2} p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) . \end{matrix}

(58)

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

\begin{matrix} p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) {log}_{2} p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) = \\ p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) {log}_{2} p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}), \end{matrix}

(59)

and

\begin{matrix} p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) {log}_{2} p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) = \\ p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) {log}_{2} p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) . \end{matrix}

(60)

Then, necessarily we have that

p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) = p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}),

(61)

and

p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) = p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) .

(62)

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

N [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] = N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] = 0.

(63)

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

n G^{T} = 0.

(64)

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

\begin{array}{l} Λ_{11} & = & 〈 ξ_{1}^{2} 〉 - {〈 ξ_{1} 〉}^{2} \\ = & \int d ξ_{1} ξ_{1}^{2} p (ξ_{1}) - {(\int d ξ_{1} ξ_{1} p (ξ_{1}))}^{2}, \end{array}

(65)

\begin{array}{l} Λ_{22} & = & 〈 ξ_{2}^{2} 〉 - {〈 ξ_{2} 〉}^{2} \\ = & \int d ξ_{2} ξ_{2}^{2} p (ξ_{2}) - {(\int d ξ_{2} ξ_{2} p (ξ_{2}))}^{2}, \end{array}

(66)

\begin{array}{l} Λ_{12} & = & 〈 ξ_{1} ξ_{2} 〉 - 〈 ξ_{1} 〉 〈 ξ_{2} 〉 \\ = & \int d ξ_{1} ξ_{1} p (ξ_{1}) \int d ξ_{2} ξ_{2} p (ξ_{2}) - \int d ξ_{1} ξ_{1} p (ξ_{1}) \int d ξ_{2} ξ_{2} p (ξ_{2}) \\ = & 0, \end{array}

(67)

Λ_{21} = Λ_{12} = 0,

(68)

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

\begin{array}{l} p^{\tilde{G}} (ξ_{1}, ξ_{2}) & = & \frac{1}{2 π \sqrt{det Λ}} exp [- \frac{1}{2} (ξ - μ) Λ^{- 1} {(ξ - μ)}^{T}] \\ = & \frac{e^{- \frac{{(ξ_{1} - 〈 ξ_{1} 〉)}^{2}}{2 Λ_{11}}}}{\sqrt{2 π Λ_{11}}} \times \frac{e^{- \frac{{(ξ_{2} - 〈 ξ_{2} 〉)}^{2}}{2 Λ_{22}}}}{\sqrt{2 π Λ_{22}}} \\ = & p^{\tilde{G}} (ξ_{1}) p^{\tilde{G}} (ξ_{2}), \end{array}

(69)

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

H [p (ξ_{1}, ξ_{2})] = H [p (ξ_{1})] + H [p (ξ_{2})],

(70)

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

\begin{array}{l} N [p (ξ_{1}, ξ_{2})] & = & H [p^{\tilde{G}} (ξ_{1}, ξ_{2})] - H [p (ξ_{1}, ξ_{2})] \\ = & H [p^{\tilde{G}} (ξ_{1})] - H [p (ξ_{1})] + H [p^{\tilde{G}} (ξ_{2})] - H [p (ξ_{2})] \\ = & N [p (ξ_{1})] + N [p (ξ_{2})] . \end{array}

(71)

If the two-mode spatial state of SPDC is a product state, the near- and far-field probability density functions are written as

p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}) = p_{F F}^{S} ({\tilde{q}}_{1}) p_{F F}^{S} ({\tilde{q}}_{2})

and

p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) = p_{N F}^{S} ({\tilde{x}}_{1}) p_{N F}^{S} ({\tilde{x}}_{2})

. Therefore, the properties mentioned above hold. Then, the total non-Gaussianity of the spatial two-photon state of SPDC will be given by

\begin{array}{l} n G^{T} & = & N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] + N [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] \\ = & N [p_{F F}^{S} ({\tilde{q}}_{1})] + N [p_{F F}^{S} ({\tilde{q}}_{2})] + N [p_{N F}^{S} ({\tilde{x}}_{1})] + N [p_{N F}^{S} ({\tilde{x}}_{2})] \\ = & n G^{M_{1}} + n G^{M_{2}}, \end{array}

(72)

where we have used that

n G^{M_{1}} \equiv N [p_{F F}^{S} ({\tilde{q}}_{1})] + N [p_{N F}^{S} ({\tilde{x}}_{1})]

and

n G^{M_{2}} \equiv N [p_{F F}^{S} ({\tilde{q}}_{2})] + N [p_{N F}^{S} ({\tilde{x}}_{2})]

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

n G^{T} = 2 \times n G^{M},

(73)

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

p_{F F}^{S} ({\tilde{q}}_{2}) = \int d {\tilde{q}}_{1} p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2}),

(74)

and

p_{N F}^{S} ({\tilde{x}}_{2}) = \int d {\tilde{x}}_{1} p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2}) .

(75)

Doing the following change of variables:

u = \frac{P}{2} ({\tilde{q}}_{1} - {\tilde{q}}_{2})

and

v = \frac{{\tilde{x}}_{1} - {\tilde{x}}_{2}}{2 P}

, we obtain that

p_{F F}^{S} ({\tilde{q}}_{2}) \propto \int d u e^{- \frac{2}{P^{2}} {(u + P {\tilde{q}}_{2})}^{2}} {sinc}^{2} u^{2},

(76)

and

p_{N F}^{S} ({\tilde{x}}_{2}) \propto \int d v e^{- 2 P^{2} {(v + \frac{{\tilde{x}}_{2}}{P})}^{2}} {sint}^{2} v^{2} .

(77)

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

p_{F F}^{S} ({\tilde{q}}_{2}) \propto \int d u δ (u + P {\tilde{q}}_{2}) {sinc}^{2} u^{2} \propto {sinc}^{2} (P^{2} {\tilde{q}}_{2}),

(78)

and

p_{N F}^{S} ({\tilde{x}}_{2}) \propto e^{- 2 {\tilde{x}}_{2}^{2}} .

(79)

In such case, it is clear that the negentropy of

p_{N F}^{S} ({\tilde{x}}_{2})

is null

[N [p_{N F}^{S} ({\tilde{x}}_{2})] = 0]

. The negentropy of the far-field marginal distribution will be the same of the function sinc²(P²q̃₂). It gives that

N [p_{F F}^{S} ({\tilde{q}}_{2})] \approx {log}_{2} (\sqrt{\frac{27 e}{32}}) - 0.4447 = 0.154.

(80)

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

p_{F F}^{S} ({\tilde{q}}_{2}) \propto e^{- 2 {\tilde{q}}_{2}^{2}},

(81)

and

p_{N F}^{S} ({\tilde{x}}_{2}) \propto \int d v δ (v + \frac{{\tilde{x}}_{2}}{P}) {sint}^{2} v^{2} \propto {sint}^{2} \frac{{\tilde{x}}_{2}^{2}}{P^{2}} .

(82)

Now, the negentropy of

p_{F F}^{S} ({\tilde{q}}_{2})

is null

[N [p_{F F}^{S} ({\tilde{q}}_{2})] = 0]

. The negentropy of the near-field marginal distribution will be the same of the function

{sint}^{2} \frac{{\tilde{x}}_{2}^{2}}{P^{2}}

. It gives that

N [p_{N F}^{S} ({\tilde{x}}_{2})] \approx {log}_{2} (\sqrt{\frac{2 π e A_{2}}{A_{1}^{3}}}) - 0.7122 = 0.224

(83)

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_{\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.

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

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]

]. 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

V = (\begin{matrix} \frac{1}{4} (1 + \frac{4 A_{2} P^{2}}{A_{1}}) & 0 & \frac{1}{4} (1 - \frac{4 A_{2} P^{2}}{A_{1}}) & 0 \\ 0 & \frac{1}{4} (1 + \frac{3}{P^{2}}) & 0 & \frac{1}{4} (1 - \frac{3}{P^{2}}) \\ \frac{1}{4} (1 - \frac{4 A_{2} P^{2}}{A_{1}}) & 0 & \frac{1}{4} (1 + \frac{4 A_{2} P^{2}}{A_{1}}) & 0 \\ 0 & \frac{1}{4} (1 - \frac{3}{P^{2}}) & 0 & \frac{1}{4} (1 + \frac{3}{P^{2}}) \end{matrix}) .

(84)

The Gaussian state associated to this covariance matrix has a purity of μ = 0.44 [28

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

], 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:

N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] = H [p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] - H [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] .

(85)

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

\begin{array}{l} N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] & = & H [p_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] + H [p_{F F}^{\tilde{G}} ({\tilde{q}}_{2})] \\ - H [p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2})] - H [p_{F F}^{S} ({\tilde{q}}_{2})] \\ = & H [p_{F F}^{\tilde{G}} ({\tilde{q}}_{1} | {\tilde{q}}_{2})] - H [p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2})] \\ + N [p_{F F}^{S} ({\tilde{q}}_{2})] . \end{array}

(86)

Note that

p_{F F}^{\tilde{G}} ({\tilde{q}}_{1} | {\tilde{q}}_{2})

does not have the same expected value and variance of the conditional probability density function

p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2})

. Thus, in general, we can not define

H [p^{\tilde{G}} ({\tilde{q}}_{1} | {\tilde{q}}_{2})] - H [p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2})]

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

p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2} = 0) \propto e^{- \frac{{\tilde{q}}_{1}^{2}}{2}} {sinc}^{2} \frac{P^{2} {\tilde{q}}_{1}^{2}}{4} .

(87)

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

p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2} = 0) \approx \frac{1}{\sqrt{2 π}} e^{- \frac{{\tilde{q}}_{1}^{2}}{2}} .

(88)

Therefore, the differential entropy is

H [p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2} = 0)] \approx \frac{1}{2} {log}_{2} (2 π e) .

(89)

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_{F F}^{\tilde{G}} ({\tilde{q}}_{1}, {\tilde{q}}_{2})

of Eq. (31). It gives the conditional Gaussian distribution

p_{F F}^{\tilde{G}} ({\tilde{q}}_{1} | {\tilde{q}}_{2})

at q̃₂ = 0 equal to

p_{F F}^{\tilde{G}} ({\tilde{q}}_{1} | {\tilde{q}}_{2} = 0) = \frac{\sqrt{3 + P^{2}}}{\sqrt{6 π}} e^{- {\tilde{q}}_{1}^{2} \frac{3 + P^{2}}{6}} .

(90)

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_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2} = 0)

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

p_{F F}^{\tilde{G}} ({\tilde{q}}_{1} | {\tilde{q}}_{2} = 0)

, such that

N [p_{F F}^{S} ({\tilde{q}}_{1}, {\tilde{q}}_{2})] = N [p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2})] + N [p_{F F}^{S} ({\tilde{q}}_{2})] .

(91)

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

p_{N F}^{S} ({\tilde{x}}_{1} | {\tilde{x}}_{2} = 0) \approx \frac{1}{2 P A_{1}} {sint}^{2} \frac{{\tilde{x}}_{1}^{2}}{4 P^{2}},

(92)

where we assumed σ = 1 for simplicity.

The variance of this function is

{({Δ {\tilde{x}}_{1} |}_{{\tilde{x}}_{2} = 0})}^{2} = \int d {\tilde{x}}_{1} {\tilde{x}}_{1}^{2} p_{N F}^{S} ({\tilde{x}}_{1} | {\tilde{x}}_{2} = 0) = \frac{4 A_{2} P^{2}}{A_{1}},

(93)

where A₁ and A₂ are the constants defined above. Now we consider the near-field Gaussian joint distribution

p_{N F}^{\tilde{G}} ({\tilde{x}}_{1}, {\tilde{x}}_{2})

. The near-field conditional distribution associated with this Gaussian approximation, when P ≪ 1, is given by

p_{N F}^{\tilde{G}} ({\tilde{x}}_{1} | {\tilde{x}}_{2} = 0) = \frac{\sqrt{A_{1}}}{2 \sqrt{2 π A_{2}} P} e^{- \frac{A_{1}}{8 A_{2} P^{2}} {\tilde{x}}_{1}^{2}} .

(94)

As one can note,

p_{N F}^{\tilde{G}} ({\tilde{x}}_{1} | {\tilde{x}}_{2} = 0)

has the same expected value and variance of

p_{N F}^{S} ({\tilde{x}}_{1} | {\tilde{x}}_{2} = 0)

. Therefore, in this limit we can write the negentropy of the near-field joint distribution as

N [p_{N F}^{S} ({\tilde{x}}_{1}, {\tilde{x}}_{2})] = N [p_{N F}^{S} ({\tilde{x}}_{1} | {\tilde{x}}_{2})] + N [p_{N F}^{S} ({\tilde{x}}_{2})],

(95)

where

N [p_{N F}^{S} ({\tilde{x}}_{1} | {\tilde{x}}_{2})]

is defined in terms of the differential entropy of

p_{N F}^{\tilde{G}} ({\tilde{x}}_{1} | {\tilde{x}}_{2})

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

n G^{T} \approx n G^{C} + n G^{M} (iff P ≪ 1),

(96)

where

n G^{C} = N [p_{N F}^{S} ({\tilde{x}}_{1} | {\tilde{x}}_{2})] + N [p_{F F}^{S} ({\tilde{q}}_{1} | {\tilde{q}}_{2})]

, and

n G^{M} = N [p_{F F}^{S} ({\tilde{q}}_{j})] + N [p_{N F}^{S} ({\tilde{x}}_{j})]

, 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.

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. 92, 210403 (2004). [CrossRef] [PubMed]
2.	M. D’Angelo, Y. H. Kim, S. P. Kulik, and Y. Shih, “Identifying entanglement using quantum ghost interference and imaging,” Phys. Rev. Lett. 92, 233601 (2004). [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]
4.	E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. 82, 2868 (1999). [CrossRef]
5.	T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. 99, 170408 (2007). [CrossRef] [PubMed]
6.	S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87 (2010). [CrossRef]
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 52, R3429 (1995). [CrossRef] [PubMed]
8.	A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett. 90, 133603 (2003). [CrossRef] [PubMed]
9.	L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett. 94, 100501 (2005). [CrossRef] [PubMed]
10.	M. N. O’Sullivan-Hale, I. Ali Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett. 94, 220501 (2005). [CrossRef]
11.	A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett. 104, 010501 (2010). [CrossRef] [PubMed]
12.	C. K. Hong and L. Mandel, “Theory of parametric frequency down conversion of light,” Phys. Rev. A 31, 2409 (1985). [CrossRef] [PubMed]
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 50 5122 (1994). [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]
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]
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]
17.	K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement migration in Hilbert space,” Phys. Rev. A 75, 050101 (2007). [CrossRef]
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]
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]
20.	D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, and F. Toscano, “Detection of transverse entanglement in phase space,” Phys. Rev. A 78, 010304 (2008). [CrossRef]
21.	D. S. Tasca, S. P. Walborn, P. H. S. Ribeiro, F. Toscano, and P. Pellat-Finet, “Propagation of transverse intensity correlations of a two-photon state,” Phys. Rev. A 79, 033801 (2009). [CrossRef]
22.	L. J. Zhang, L. Neves, J. S. Lundeen, and I. A. Walmsley, “A characterization of the single-photon sensitivity of an electron multiplying charge-coupled device,” J. Phys. B 42, 114011 (2009). [CrossRef]
23.	H. Di Lorenzo Pires and M. P. van Exter, “Observation of near-field correlations in spontaneous parametric down-conversion,” Phys. Rev. A 79, 041801 (2009). [CrossRef]
24.	M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: the Einstein-Podolsky-Rosen paradox: from concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009). [CrossRef]
25.	R. M. Gomes, A. Salles, F. Toscano, P. H. S. Ribeiro, and S. P. Walborn, “Quantum entanglement beyond Gaussian criteria,” Proc. Natl. Acad. Sci. U.S.A. 106, 21517 (2009). [CrossRef] [PubMed]
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]
30.	H. Di Lorenzo Pires and M. P. van Exter, “Near-field correlations in the two-photon field,” Phys. Rev. A 80, 053820 (2009). [CrossRef]
31.	S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett. 88, 120401 (2002). [CrossRef] [PubMed]
32.	T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991). [CrossRef]
33.	P. Comon, “Independent component analysis, A new concept?,” Sig. Process. 36, 287–314 (1994). [CrossRef]
34.	P. B. Dixon, G. A. Howland, J. Schneeloch, and J. C. Howell, “Quantum mutual information capacity for high dimensional entangled states,” arXiv:1107.5245v1[quant-ph].
35.	M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett. 96, 080502 (2006). [CrossRef] [PubMed]

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

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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]

Abouraddy, A. F.
Aguirre Gómez, J. G.
Ali Khan, I.
Andersen, U. L.
Bachor, H. A.
Banaszek, K.
Barnett, S. M.
Bennink, R. S.
Bentley, S. J.
Bobrov, I. B.
Bowen, W. P.
Boyd, R.
Boyd, R. W.
Brambilla, E.
Cavalcanti, E. G.
Chan, K. W.
Cirac, J. I.
Comon, P.
Cover, T. M.
D’Angelo, M.
de Matos Filho, R. L.
Di Lorenzo Pires, H.
Dixon, P. B.
Drummond, P. D.
Eberly, J. H.
Einstein, A.
Eisert, J.
Ether, D. S.
Fonseca, E. J. S.
Franke-Arnold, S.
Gatti, A.
Genoni, M. G.
Giedke, G.
Giovannetti, V.
Gomes, R. M.
Henkel, C.
Hong, C. K.
Howell, J. C.
Howland, G. A.
Hyvärinen, A.
Ivanov, D. P.
Jack, B.
Jha, A. K.
Kalinkin, A. A.
Karhunen, J.
Kim, Y. H.
Klyshko, D. N.
Korn, D.
Kulik, S. P.
Lam, P. K.
Law, C. K.
Leach, J.
Leuchs, G.
Lima, G.
Lugiato, L. A.
Lundeen, J. S.
Mancini, S.
Mandel, L.
Monken, C. H.
Neves, L.
O’Sullivan-Hale, M. N.
Oja, E.
Ostermeyer, M.
Padgett, M. J.
Pádua, S.
Paris, M. G. A.
Pellat-Finet, P.
Pittman, T. B.
Podolsky, B.
Puhlmann, D.
Reid, M. D.
Ribeiro, P. H. S.
Rosen, N.
Rubin, M. H.
Saavedra, C.
Saleh, B. E. A.
Salles, A.
Schneeloch, J.
Sergienko, A. V.
Shih, Y.
Shih, Y. H.
Straupe, S. S.
Strekalov, D. V.
Tasca, D. S.
Teich, M. C.
Thomas, J. A.
Tombesi, P.
Torres, J. P.
Toscano, F.
van Exter, M. P.
Vitali, D.
Walborn, S. P.
Walmsley, I. A.
Wolf, M. M.
Yarnall, T.
Zagury, N.
Zhang, L. J.

J. Mod. Opt.

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

J. Phys. B

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]

Phys. Rep.

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

Phys. Rev.

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

Phys. Rev. A

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]
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]
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]
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]
H. Di Lorenzo Pires and M. P. van Exter, “Near-field correlations in the two-photon field,” Phys. Rev. A80, 053820 (2009). [CrossRef]
M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian character of a quantum state by quantum relative entropy,” Phys. Rev. A78, 060303 (2008). [CrossRef]
M. G. Genoni and M. G. A. Paris, “Quantifying non-Gaussianity for quantum information,” Phys. Rev. A82, 052341 (2010). [CrossRef]

Phys. Rev. Lett.

M. M. Wolf, G. Giedke, and J. I. Cirac, “Extremality of Gaussian quantum states,” Phys. Rev. Lett.96, 080502 (2006). [CrossRef] [PubMed]
S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling macroscopic oscillators exploiting radiation pressure,” Phys. Rev. Lett.88, 120401 (2002). [CrossRef] [PubMed]
C. K. Law and J. H. Eberly, “Analysis and Interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett.92, 127903 (2004). [CrossRef] [PubMed]
A. Gatti, E. Brambilla, and L. A. Lugiato, “Entangled imaging and wave-particle duality: from the microscopic to the macroscopic realm,” Phys. Rev. Lett.90, 133603 (2003). [CrossRef] [PubMed]
L. Neves, G. Lima, J. G. Aguirre Gómez, C. H. Monken, C. Saavedra, and S. Pádua, “Generation of entangled states of qudits using twin photons,” Phys. Rev. Lett.94, 100501 (2005). [CrossRef] [PubMed]
M. N. O’Sullivan-Hale, I. Ali Khan, R. W. Boyd, and J. C. Howell, “Pixel entanglement: experimental realization of optically entangled d = 3 and d = 6 qudits,” Phys. Rev. Lett.94, 220501 (2005). [CrossRef]
A. K. Jha, J. Leach, B. Jack, S. Franke-Arnold, S. M. Barnett, R. Boyd, and M. J. Padgett, “Angular two-photon interference and angular two-qubit states,” Phys. Rev. Lett.104, 010501 (2010). [CrossRef] [PubMed]
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]
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]

Proc. Natl. Acad. Sci. U.S.A.

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

Rev. Mod. Phys.

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]

Sig. Process.

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

Other

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].
A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001). [CrossRef]
T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991). [CrossRef]

2011, Straupe, Phys. Rev. A
2010, Genoni, Phys. Rev. A
2010, Walborn, Phys. Rep.
2010, Jha, Phys. Rev. Lett.
2009, Di Lorenzo Pires, Phys. Rev. A
2009, Ostermeyer, J. Mod. Opt.
2009, Di Lorenzo Pires, Phys. Rev. A
2009, Tasca, Phys. Rev. A
2009, Zhang, J. Phys. B
2009, Di Lorenzo Pires, Phys. Rev. A
2009, Reid, Rev. Mod. Phys.
2009, Gomes, Proc. Natl. Acad. Sci. U.S.A.
2008, Genoni, Phys. Rev. A
2008, Tasca, Phys. Rev. A
2007, Walborn, Phys. Rev. A
2007, Chan, Phys. Rev. A
2007, Yarnall, Phys. Rev. Lett.
2006, Wolf, Phys. Rev. Lett.
2005, Neves, Phys. Rev. Lett.
2005, O’Sullivan-Hale, Phys. Rev. Lett.
2004, Howell, Phys. Rev. Lett.
2004, D’Angelo, Phys. Rev. Lett.
2004, Law, Phys. Rev. Lett.
2003, Gatti, Phys. Rev. Lett.
2002, Mancini, Phys. Rev. Lett.
1999, Fonseca, Phys. Rev. Lett.
1998, Monken, Phys. Rev. A
1995, Pittman, Phys. Rev. A
1994, Rubin, Phys. Rev. A
1994, Comon, Sig. Process.
1985, Hong, Phys. Rev. A
1935, Einstein, Phys. Rev.

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