Imaging high-dimensional spatial entanglement with a camera. M. Edgar, D. Tasca, F. Izdebski, R. Warburton, J. Leach, M. Agnew, G. Buller, R. Boyd and M. Padgett in Nature Comm. 3:984 (2012).  What the paper says!?

This shows the spatial (near-field) and momentum (far-field) imaging of correlations from a SPDC source with a 201×201 array of pixels camera:

Here we report the observation of spatial correlations from SPDC using an EMCCD camera in both the image plane and far field of the nonlinear crystal showing position correlation and momentum anti-correlation, respectively.

As such, this implements in a single-shot measurement the earlier results from Howell et al., which were scanned by hand, but measuring both spaces, while the earlier still spatial measurements by Jost et al. with a camera were for the momentum space only. This paper ties everything together. They also link it to EPR measurements, while recognizing various limitations (background, locality of the camera itself, etc.) They add a dimension of high-multidimensional entanglement:

In both bases, we show correlations of around 50 modes in both transverse directions, giving in principle access to several thousand entangled spatial states

and in more details:

The joint detection probability is $\mathcal{P}(\boldsymbol{\rho}_1,\boldsymbol{\rho}_2)\propto\vert\Psi(\boldsymbol{\rho}_1,\boldsymbol{\rho}_2)\vert^2$. According to the Gaussian model and the parameters of our system, we predict the number of modes for joint detections in both position and momentum to be $(\sigma_+/\sigma_-)^2$≈3,500.

Although not yet exploited, this also sets a record:

this represents the largest dimensionality for any experiment using entangled spatial states of photons.

They suffer a lot from so-called "charge smearing", which they don't explain, just lament. This is in the $y$ direction only, making them discard their spatial correlation along this axis (Fig. 4b) although in space this doesn't differ too much from the x-case (Fig. 4a) except for stronger correlations. The clue is however in the momentum space, where they see a clear unwanted diagonal in $y$ on top of the anti-correlation antidiagonal.

Overall, the paper does well what other have done before, in particular confirming EPR violation. There is little if anything that is conceptually new.

The Gaussian spatial structure of their biphoton is given as:

This is known as « the transverse wave function of the post-selected two-photon field of SPDC for a Gaussian pump beam» and they cite Ref.  for that. Type 1 phase matching reults in no spatial separation between the twin photons.

There is a nice tension between discrete & continuous bases with the high dimensionality of the transverse spatial degrees of freedom:

The high dimensionality of the transverse spatial degrees of freedom (DOF) of the photon pairs can be explored either by projections onto a discrete basis, such as the Laguerre–Gaussian modes, or by using a continuous basis defined by the transverse position or momentum of the photons

And a nice motivation for the necessity of cameras:

This sequential scanning or use of a small number of detectors negates any information capacity advantage in the use of spatial states. In all protocols for highdimensional quantum key distribution, quantum computation and teleportation using spatial states, it is essential to perform a full field measurement of the photon transverse position, which is made possible by using a two-dimensional (2D) detector array.