Eric,
"This quickly got into the measurement problem." Because that is the only problem.
"I wish to steer clear of interpreting what the measurement is telling us about the system before the measurement." There is no other alternative, precisely because it has turned out that quantum theory has never even attempted to describe the things being measured. It is a strictly a theory that only describes the act of measurement itself. The "paradoxical or incoherent" nature arises entirely from the false assumption, that the theory ought to be describing the things being measured. The "shut up and calculate" school of thought, arose precisely to discourage students from making any such assumption; a message that has been lost on the last generation of physicists, resulting in the "industry standard" teachings, becoming deeply flawed. The entire, original point of the cat and EPR paradoxes, was an attempt to convince people that something extremely fundamental MUST be misunderstood, about what the mathematics is actually describing. That something has turned out to be the behavior, DURING THE MEASUREMENT PROCESS, of a single, classical bit of information, as described by Shannon; it behaves like a mythical qubit.
In effect a "phase change" occurs, like water freezing into ice, when the information content of an observable shrinks from more than one bit, to only a single bit, because it is then impossible, even in principle, to make two uncorrelated measurements of such a single, classical bit. That is why the Heisenberg uncertainty principle exists. And that is why the seemingly strange correlations appear, whenever two measurements are attempted on such an entity. Variables, like position and momentum, that were formerly independent, can no longer be independent, once that "phase change" has occurred, and the variables have become "frozen together", rather like two molecules of water becoming frozen together and consequently unable to move independently.
It is important to realize that none of this has anything directly to do with physics or its interpretation: it is a purely mathematical property of information itself. Shannon's Capacity Theorem, is a theorem, not a theory - a statement of a mathematical relationship between continuous functions and the discrete measurements of any such functions. It does not matter in the slightest, that physicists have assigned some physical "meaning" or "interpretation" to the various functions they employ; such as THAT variable represents "position" and THAT one represents "momentum". What the functions are deemed to represent is irrelevant. The problem lies within the fundamental, mathematical nature of a function itself - the property of being bandlimited.
Rob McEachern