Realization of the Einstein-Podolsky-Rosen Paradox Using Momentum- and Position-Entangled Photons from Spontaneous Parametric Down Conversion. J. Howell, R. Bennink, S. Bentley and R. Boyd in Phys. Rev. Lett. 92:210403 (2004).  What the paper says!?

This is a seminal realization of the EPR paradox as initially envisioned by E, P & R:

Here we report on a demonstration of the EPR paradox using position- and momentum-entangled photon pairs produced by spontaneous parametric down conversion.

It is implemented with a SPDC source:

the idealized entangled state proposed by EPR [...] is non-normalizable and cannot be realized in the laboratory. However, the state of the light produced in parametric down conversion can be made to approximate the EPR state

The theory of transverse entanglement is attributed to Law and Eberly (no mention of Klyshko). Unlike previous works by Malygin et al. and Jost et al., here "real" spatial imaging of the SPDC is performed. The photon pair is however created at the same point by the source and is imaged straight from the crystal exit face. So there is no spatial structure either, unlike our case. The point is that since they also measure (like in previous works) the momentum correlations, i.e., they measure correlations in both spaces, they can realize the EPR experiment, following an idea from Gatti et al.:

The idea is to measure the positions and momenta by measuring the down converted photons in the near and far fields, respectively

They report a joint conditional uncertainty that is essentially zero:

but this does not falsify Heisenberg uncertainty because $x_1$ and $p_1$ are not jointly measurable anyway, so the two conditional variances come from mutually exclusive experiments, while photon 2 on its own remains a broad mixed state, with $\Delta x_2\,\Delta p_2\gg\hbar/2$. This is evaluated at a single $x_1$ but remains broadly independent of which $x_1$ they take (if they trim off the tails, they even improve a lot the situation).

They also use an entanglement witness (see its derivation): $$\mathrm{Var}(\hat x_1-\hat x_2)\;\mathrm{Var}(\hat p_1+\hat p_2)\;\geq\;\hbar^{2}\quad\text{for all separable }\rho$$ which they pulverise, making compelling their entanglement. This criterion (by Mancini et al.) seems the most convenient (if not best available) for this type of EPR violation.

The near-field imaging is made with a f=100mm lens before the PBS which projects the exit face of the crystal onto two slit planes, where each photon is retrieved: slit 1 is parked at peak singles rate while slit 2 rides a translation stage. Coincidence rate versus slit-2 displacement provides $P(x_2|x_1)$.

Similarly, for far-field imaging, the above lens is removed and replaced by two lenses in each arm, a focal length from the slits. Now each slit plane is a Fourier plane $x = fk_\perp/k$. In summary:

Near- vs far-field correlations in the two detection configurations.

	Frame-by-frame	Joint structure
Near field	Dot in arm 1 at random x, dot in arm 2 at nearly the same x	$x_1-x_2$ sharp $(\sim 27\mu \mathrm{m})$
Far field	Dot in arm 1 at random x, dot in arm 2 at nearly −x	$p_1+p_2$ sharp $(\sim 3.7\hslash /\mathrm{m}\mathrm{m})$

Averaging all frames would produce the Gaussian spot for the near-field, and the phase-matching pattern for the far-field (a ring for noncollinear phase-matching, a lobe for collinear as is the case here).

The whole discussion on EPR is interesting:

The standard view is that entangled particles interact nonlocally, in contradiction to EPR’s assumption.

Entanglement and nonlocality:

To interpret these results, it is helpful to consider the relationship between the original EPR paradox and the issues of entanglement, nonlocality, and quantum signatures, which have been the subjects of more modern studies. The intent of EPR was not to reveal a discrepancy between classical and quantum theory, but to show that quantum mechanics is ‘‘incomplete’’ in the sense that noncommuting observables such as x₂ and p₂ could be known with more certainty than is allowed by the uncertainty principle

On elements of reality:

In the situation proposed by EPR, the position or momentum of the unmeasured particle becomes a reality when, and only when, the corresponding quantity of the other particle is measured. Since only one quantity or the other is measured, the position and the momentum of the unmeasured particle need not be simultaneous realities. In this way the paradox is resolved.

They discuss a related Popper experiment.

What other people say

Moreau et al. describes this paper as such ("they did not prospect the full EPR characteristic"):

Howell et al. [8] have measured in both planes the probability distribution of the position of photon 2, conditioned by the detection of photon 1. The product of the conditional variances was found to be 25 times smaller than the limit for the product of variances for a single photon given by Heisenberg’s uncertainty relation. This impressive result was obtained by measuring temporal coincidences between cross-polarized photons in type-2 SPDC. These photons were separated by a polarizing beam splitter: for a fixed position of a narrow slit transmitting photon 1 to an avalanche photodiode, the level of coincidences was measured for each position of a similar slit transmitting photon 2 to a separate similar detector. Hence, in the words of Reid et al. [5], “detection events are only considered if two emitted photons are simultaneously detected.” In this sense, they did not prospect the full EPR characteristic of SPDC but a monodimensional and point EPR paradox using postselected data.

And this is comes from Lantz et al.:

Howell et al [2] measured in both planes the probability distribution of the position of the second photon, conditioned by the detection of the first photon behind a slit. This type of detection selects a priori photons experiencing temporal coincidences.

They proceed to say that this was different in their own works.