  The state of the spatially correlated down-converted photons is usually
treated as a two-mode Gaussian entangled state. While intuitively this seems to
be reasonable, it is known that new structures in the spatial distributions of
these photons can be observed when the phase-matching conditions are properly
taken into account. Here, we study how the variances of the near- and far-field
conditional probabilities are affected by the phase-matching functions, and we
analyze the role of the EPR-criterion regarding the non-Gaussianity and
entanglement detection of the spatial two-photon state of spontaneous
parametric down-conversion (SPDC). Then we introduce a statistical measure,
based on the negentropy of the joint distributions at the near- and far-field
planes, which allows for the quantification of the non-Gaussianity of this
state. This measure of non-Gaussianity requires only the measurement of the
autocorrelation covariance sub-matrices, and will be relevant for new
applications of the spatial correlation of SPDC in CV quantum information
processing.
