For example, take quantum decoherence (which, IMHO, is the most logical explanation for the collapse of the wave-function - by saying it does not actually collapse). Quantum decoherence is almost like a giant constraint resolution system - once a particle randomly interacts with another they become entangled and both now have fewer degrees of freedom. When it interacts with many particles, like any macro-effect it has essentially no degrees of freedom anymore. It's all about who knew about what and when. The experiments around this fascinating. (Note that there are other theories, like the many-worlds interpretation, that also explain the collapse of the wave function)
This seems to be another example of this. Anyway, as I said, just a layman.
In the same circumstances a light beam stops producing an interference pattern in the Young experiment, quantum wave functions do as well. This is pretty easy to derive, just introduce a random phase shift term, and average across it, and the interference pattern disappears and a bell curve emerges instead.
A state is entangled when it isn’t a product state.
Two spin (1/2) particles in a singlet state have the kind of “they have opposite states” thing going on that you describe, and is a specific way that two particles can be entangled.
As a non-mathematician this is a wonderfully evocative summary. I'm curious if mathematicians find it a reasonable characterization of the proof.