Realization of the purely spatial Einstein-Podolsky-Rosen paradox in full-field images of spontaneous parametric down-conversion. P. Moreau, J. Mougin-Sisini, F. Devaux and E. Lantz in Phys. Rev. A 86:010101 (2012).  What the paper says!?

This paper has «demonstrated a purely spatial EPR paradox by using a full-field and direct detection method.» This follows up on previous works such as Howell et al. of showing that the Heisenberg uncertainty $\Delta x^2\Delta p_x^2\ge{\hbar^2\over4}$ can be violated thanks to entangled pairs. Here they do it for the full field of the camera:

By recording all the photon pairs generated by spontaneous parametric down-conversion in the near field and the far field of the same system, we make fewer supplementary assumptions than previous papers that involve postselection or homodyne detection. Hence, this demonstration of one of the most fascinating phenomena of quantum mechanics is made in the form closest to its original formulation.

They report violation of

Their interpretation of EPR:

quantum mechanics predicts that these particles could have both perfectly correlated positions and momenta, in contradiction to the so-called local realism where two distant particles should be treated as two different systems. Though the original intention of EPR was to show that quantum mechanics is not complete, the standard present view is that entangled particles do experience nonlocal correlations

SPDC at it again:

Spontaneous parametric down-conversion (SPDC) provides independent pairs of entangled photons that makes the system very close to that considered in the original EPR paper: the positions of photons 1 and 2 are detected in the near field, and their momenta correspond to the far field.

Type-2 separates spatially the signal from the idler:

the use of type-2 phase matching allows us to spatially separate the idler and signal photons to be close to the real conditions of an EPR test of local realism.

They "criticize" earlier works for not mapping the full EPR panorama but postselecting to a single point, and here:

We present here a full spatial demonstration of EPR steering by imaging highly spatially multimode type-2 SPDC with an electron-multiplying CCD (EMCCD) camera. This ensures taking into account each detection event that occurs in a relatively long exposure time compared to the laser pulse duration and, more importantly, compared to the coincidence time detection used in the experiments that use correlation data detection. Hence, even if there are losses and false-positive and -negative events, these spurious events are random, which is fundamentally different from considering a priori that pairs are correlated and detecting only in the temporal and spatial gate where the twin photon arrives.

They also make an interesting statement regarding the wholeness of the system as opposed to its projected subsystems:

Second, Heisenberg inequalities concern a sole system, hence it is meaningless to test these inequalities if the near field and the far field do not correspond to this unique system. By treating full-field bidimensional images of photodetection and measuring variances in two orthogonal directions, we assure a perfect correspondence between the subsystems involved in the near field and in the far field [5], in contrast to point or one-dimensional (1D) detection.

This is one of the important inputs from this paper: instead of selecting a particular area or point on the plane, everything is included.

This is a worthwhile thing to do because that involves a very-high multimode field:

The etendue of the beam, i.e., the product of its transverse surface by the solid angle it subtends or the number of transverse modes in appropriate units, corresponds to the two-photon Schmidt number

Note that "étendue" is a valid term to describe the spreading of light.

Their description of EPR for SPDC is crystal clear:

The spatial extension of a mode in either the near or the far field is proportional to the inverse of the full beam extension in the other plane. For single-photon imaging, the laws of diffraction are equivalent to Heisenberg’s uncertainty relation: a photon that can be localized in one mode of the near field, for example, by traversing an aperture of the size corresponding to the mode, will be detected at a random position in the entire far-field diffraction pattern. However, the laws of quantum mechanics state that a pair of signal-idler photons will be detected either in the same mode in the near field or in opposite modes in the far field if no detection occurs in the other plane. Because the detection plane can be chosen at a time when causal interaction between photons is no longer possible, these correlations are not compatible with local realism, as demonstrated first in the EPR paper [1], though compatible with Heisenberg’s uncertainty relation since correlations cannot be measured in both planes for the same photon pair.

Importantly, when looking at the correlation function they derive (some variant of a $g^{(2)}$):

they discuss the type of average to be done:

The mean in this equation can be estimated by spatial averages on the different pixels of the image for a fixed $\Delta r$, given 1-2 by the intercorrelation of two “regions of interest” (ROIs) of an image, each one corresponding to one polarization of the SPDC. We will therefore obtain a nonlocal estimation involving all the light.

This is unclear to me:

Because of the weak signal-to-noise ratio, we proceed to an additional statistical average on different images taken at different times for the same system configuration.

They provide images of their near-field (left) and far-field (right) statistical averages:

From individual frames, they compute what they call "intercorrelation" functions:

Products of such quantities bring them to their central result: realization of EPR correlations over the full field, without post-selection.

There is a more detailed version of this work in Ref. .