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If every time a particle's spin or momentum is measured, it gives a discrete answer (collapse of possibility states), how can they ever prove that prior to measurement it was in fact in a super-position of states? Is this solely a logical extrapolation from the wave-like interference patterns seen in the slit experiment?

Clearly I don't understand something fundamental here.

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There is an approach called weak measurement that can be used to probe the properties of a superposition without destroying it.

There is a reasonable detailed article on it on Wikipedia, or a more accessible article on the Nature web site.

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One can prepare a lot of the same quantum states and do the same measurement to them, then he will get a series of results. From the this results one can know the possibilities of getting different results, then one can know what the initial quantum state is like. But sometime it cannot be told that whether the initial state is pure or mixed.

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If varying results is the only way we prove that the initial state was mixed, then how is it distinguishable from a measurement inaccuracy problem? –  Mike S Apr 29 at 3:49
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You question involves a misrepresentation of how science works. Science doesn't prove anything. Rather, scientists make progress by doing the following, as explained by Karl Popper (see "Realism and the Aim of Science", Chapter I). (If they don't do what I'm describing they don't make progress.) (1) They propose ideas to try to solve problems with current theories. (2) They then look for problems with their proposals, such as failing to solve the problem they were created to solve or clashing with a new experimental result. (3) Once they have a theory that solves the original problem and doesn't have any outstanding problems they start to look for problems with the new idea.

So then the question that should be asked is, if there is a superposition of states before measurement how could that idea be tested? What testable statement would have to be true if there is a superposition that would not be true if there was no superposition? There are many experiments that do such tests. If you do a single particle interference experiment in an interferometer and put a phase shifter in one arm of the interferometer that tests whether there was a superposition.

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