My basic understanding is that Superposition is a property of particles that allow them to behave as if they are in multiple states simultaneously, e.g., interference patterns in double-slit experiment. Quantum Entanglement means that if I measure a state of one of two entangled particles, I can know the corresponding state in the other. So, that begs the question: If Alice and Bob produces two buckets of entangled photons, A and B, and Bob takes bucket B and measures each of them, then Alice takes bucket A and performs a double slit experiment with them, will Alice still get interference patterns? In other words, can the particles still behave as if in a Superposition even if I know the state of each one?
This is an extreme oversimplification:
Quantum Entanglement means that if I measure a state of one of two entangled particles, I can know the corresponding state in the other.
There are many types of entanglement (or, more specifically, pairs particles can show entanglement on many different degrees of freedom), but generally speaking entanglement is a property that shows up as a kind of correlation between the two particles. For entangled particles, experiments on either of the two particles will not show any interference properties at all, and it is only once you collate results with what the other particle produced that you can obtain interesting stuff from the correlations.
Once you do collate the results, though, what you get will depend on what kind of measurement you did on each of the two particles. Generally speaking, if the measurement on B provides you with enough information to infer (via correlations on the later analysis) which slit in a double-slit experiment the A particle went through, then you will be completely unable to recover any interference in the post-slits pattern. However, there are measurements on B that can reverse this decoherence (normally known as quantum eraser experiments) where the measurement on B is incompatible (in the technical sense) with any which-slit measurement, in which case post-selection on that measurement can reveal interference patterns within the data in A.
In the usual case, the polarisation state of the photons will be entangled whereas a double-slit experiment is agnostic to polarisation. So in that most common case, Alice will still see interference patterns. However if you were to install a polariser on one of the slit, depending on the setup, you could make the interference disappear. Take this with a grain of salt: this is just to convey the gist of it as the devil is in the details.
Great question indeed.
Well, this whole topic is speculative as there can be many assumptions and uncertainties. Therefore, to avoid tangential comments from Q guards, I will take a specific scenario from your question.
Suppose the entanglement/state involves perfect anti correlation between the two particles of a pair.
Suppose the setup is such that Alice is on one side of the source and double slit is on the other side (Bob side) of the source.
Suppose Alice measures her side of the photon, just a moment before the Bob side photon enters the double slit. That moment is small enough for light speed to not being able to communicate from Alice to Bob side.
Suppose numerous entangled pairs are sent one by one in this setup.
Now per various current theories, the entanglement will be broken instantaneously on Alice's measurement.
One such theory/interpretation is that the wave function collapses. Let us consider this interpretation.
So, as the wave function collapses before the Bob side photon hits the double slit, it should not behave as a wave and there should be no interference pattern. Because the interference pattern is caused by wave nature, which has collapsed.
This setup can be used to investigate the truth behind nature of entanglement - i.e. whether the two photons are indeed a single system or not.
Obviously, depending upon whether the interference is observed in such an experiment, folks can argue that the wave function that collapsed on Alice's measurement is only the one that ties the two particles into a single system and the wave function that causes interference is a different wave function and survives the collapse of entanglement wave. However, both wave functions appear to follow same math, so I would think it should be same and the one wave.
Again, great question, and a great scenario for investigating truth of entanglement.