Can observations of entangled particles affect their unobserved counterparts? There are two experiments that are often used to explain Quantum Mechanics: the two-slit experiment and the EPR paradox.  I am curious what would happen if you combined them.
Imagine an experiment where you fire pairs of entangled particles at two simultaneous two-slit setups.  If you used detectors, you could find out how the entangled particles' paths correlate.  Perhaps you'd be able to deduce, based on the result of one detector, which slit the other particle went through.  Now, if you were to run the experiments with a detector on one side and no detector on the other side, would the unobserved particles still form an interference pattern, even though you know which slit they would have went through?
My intuition is that the answer is yes.  Despite being entangled, the particles should not have correlated actions, otherwise we would have invented faster-that-light communication.  You could create a device that constantly fired entangled particles toward two far-apart worlds, and if one world suddenly started observing the particles on their side, the particles arriving at the other world would instantly cease to create an interference pattern.
 A: An experiment very close to the proposed experiment has been done by Zeilingers grad-student Dopfler in 1998. She used a down-conversion crystal to produce pairs of entangled photons in a quantum-eraser type of experiment involving a dual-slit.
One of the members went through the dual-slit and was detected by a detector A which is scanning the space behind the slits to see if there is an interference pattern or not. The other member passes through a lens to another detector B whose distance to the lens can be varied (angle is fixed). By moving the detector B into or out of focus with the lens, the other pair member can be detected as passing through one of the slits (in focus, it "sees" the slit holes) or the which-way information is erased (out of focus, information from both slits are merged).
A useful way of looking at these experimental setups is to pretend that the photon is emitted by one of the detectors, passing backwards through the experiment, through the down-conversion crystal with momentum intact, and finally being absorbed by the other detector.
This experiment is thus described simply as the lens + detector B either watching the which-slit information or not at the other section of the experiment.
According to Dopfler and Zeilinger the experiment worked, but I have not read anything about it since and the original dissertation has been pulled from the web, but a copy can be found at the internet archive. Since it used a coincidence counter to increase the signal-to-noise between both detectors, they had not really demonstrated any FTL-signalling, however the speculation is that the experiment could run without the counter. Zeilinger calls the conceptual alternative to spacelike signalling "retrocausality" I think, where the cause is timelike in both paths after the down-conversion, but runs backwards in one of them (like in the "pretend" tool mentioned above).
J. Cramer apparently currently works on refining this experiment.
A: There are a number of papers that study double slit interference at the single photon level using entangled photon pairs. The paper by Scarcelli et al. https://arxiv.org/abs/quant-ph/0512207 conducts a delayed choice quantum eraser experiment using a ghost imaging set up to show classic interference fringes when which-path information is absent, and the absence of interference when which path information is present. Overall, the interference is present in the data as a subset of all possible detections through the use of a Fourier dc band-pass filter. In contrast, when all possible coincidences are collected there is no interference. That is odd! 
In contrast a similar experiment, published by Reubin et al. https://arxiv.org/abs/1602.05987, differing only in the absence of the ghost imaging lens reveals the classic interference pattern for all possible coincidences. However with one crucial difference. They had to use optical mode filtering to achieve high quality interference.
In both cases, one photon is simply passed through the double slit and detected. 
These experiments are both single photon double slit interference experiments. However they reveal one oft overlooked aspect to conducting such experiments. The nature of a single photon.
Single photons are single excitations of an electromagnetic field mode. As a quantum particle they can exist in superpositions. As such they can also represent a multimode state.
When we are looking at double slit interference, it is often implicit that we are dealing with single mode excitation, with a well-defined phase at the slits. This is true up the the macroscopic limit with a coherent single mode laser. The importance of the mode composition of the illuminating beam is crucially important for observing interference. This is always a part of any undergraduate experiment of the double slit.
However, mode composition seems to be forgotten at the single photon level.
Experiments conducted using entangled photons pairs often neglect the mode quality of the beam. This was illustrated in the paper by Reubin et al., where they showed that mode filtering was necessary to achieve good interference fringes. 
The paper that purports to illustrate the delayed choice quantum eraser, by Scarcelli et al., also employs mode filtering in their experiment using variable with windows in the Fourier transform plane. 
What is not considered is the effect of filtering on the entangled partner. The two photons emitted from the down conversion crystal are momentum entangled. That means they are effectively mode entangled. When you apply a mode filter to one photon on one arm, you are restricting the interference pattern to only those photons with the same spatial mode. 
Thus the purported delayed choice quantum eraser experiments are in reality just illustrating post-selective mode filtering. The "which path" information is irrelevant in these experiments because the collapse of the wavefunction still occurs at the detector or CCD in all cases. It is just that the experiment has the ability to choose different wavefunctions using the photon correlated mode filtering technique described here, where the broad featureless pattern is easily interpreted as the probability distribution corresponding to multimode excitation. 
Thus, these experiments demonstrate that entangled photon mode selection can indeed affect the experimental outcome of a double slit experiment. The results are however closely related to the ghost imaging results demonstrated in the paper by Reubin et al., and otherwise do not conclusively demonstrate the delayed choice quantum eraser effect.
