can entangled photons produce an interference pattern? Suppose you have a laser that emits entangled photons in opposite directions towards a double-slit screen setup as shown below.

The photons are entangled such that if a photon on the left travels through slit A, then its entangled photon goes through slit D.  If we do not record any which path information on either side, would we get an interference or clump pattern recorded on the screens beyond the double-slits?
 A: You can infer what happens by looking at one of the screens only (say, the left one). In that case, instead of entangled photons, you can as well just look at the left photons.  If some of their degrees of freedom (DoF) are entangled with the right photons, it means that the left photons don't have a well-defined value for that DoF (i.e. it is in a mixed state).  So the question is whether you still see an interference pattern if some degree of freedom has some randomness.
First, note that every photon just interferes with itself. Thus, the distribution of each individual photon on the screen will follow an interference pattern.
Now if the entangled DoF is the polarization, each photon will follow the same interference pattern, and you will see an interference pattern.
On the other hand, if the entangled DoF is the frequency, the spacing of the pattern will be different for each photon, and thus, you will see the average of interference patterns over the spread of the frequency.  For a large enough spread, this will destroy the interference pattern (the spacing is proportional to the wavelength).  If you want to know how this changes with the spreading of the frequency, this is  a nice exercise problem!
