If I had a pair of entangled particles that were both in superposition, and I put one of them through a interference experiment, such as the double slit experiment, would I see an absence of interference the moment I measure the other particle? And how would this change if both were in an interference experiment and I measure one of them?
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Interestingly: When two particles - say photons - are entangled, neither will produce double slit interference at all. (For them to produce a double slit pattern, they must not be entangled and must be made coherent.) Measuring one of them will not cause any detectable/observable change in the double slit pattern of the other on its own. This is true for any entangled particle pairs.
Were this not the case, it would be possible to send signals faster than light (FTL). See figure 2 of the following article by Zeilinger:
http://courses.washington.edu/ega/more_papers/zeilinger.pdf
There are experimental cases in which double slit interference patterns will appear for two entangled photons when coincidences are recorded (i.e. observation of outcomes on both particles are compared). However, you can't send an FTL signal in this manner, as classical communication is required to obtain the coincidence information.
