I recently misread this question and wrote a nice answer for what turned out not to be the question at all. I then proceeded to look quite carefully for the question I had actually answered, and I found that we don't really have a question (or answers) about the specific scheme described in
- The choice of measurement basis on one half of an entangled state affects the other half. Can this be used to communicate faster than light?
I therefore went ahead and posted it as a self-answered Q&A.
I know we have had a ton of similar questions, and we have certainly touched on it edgewise (as in e.g. this answer), but I could not find this specific scheme, which is otherwise quite a popular misconception about entanglement.
The closest related question, I think, is EPR-type experiments and faster-than-light communication using interference effects as signaling mechanism, but this question (a) uses collapse and not measurement as a proposed method, and (b) is unclear and not particularly helpful for future visitors.
This thread does not serve any particular purpose, but I set it up to avoid long discussions in comments. If you have proposed duplicates, I would be interested in seeing them and discussing whether they are indeed or not. I do feel, though, that we need a nice, simple and understandable canonical question for this scheme.