[epistemic status: I'm a mathematical and philosophical expert but not a QM expert; conclusions are very much tentative]
There is tension between the following two claims:
- The fundamental nature of reality consists of the wave function whose evolution follows the Schrödinger equation.
- Some discrete facts are known.
(What is discrete knowledge? It is knowledge that some nontrivial proposition X is definitely true. The sort of knowledge a Bayesian may update on, and the sort of knowledge that logic applies to.)
The issue here is that facts are facts about something. If quantum mechanics has any epistemic basis, then at least some things are known, e.g. the words in a book on quantum mechanics, or the outcomes of QM experiments. The question is what this knowledge is about.
If the fundamental nature of reality is the wave function, then these facts must be facts about the wave function. But, this runs into problems.
Suppose the fact in question is "A photon passed through the measurement apparatus". How does this translate to a fact about the wave function?
The wave function consists of a mapping from the configuration space (some subset of R^n) to complex numbers. Some configurations (R^n points) have a photon at a given location and some don't. So the fact of a photon passing through the apparatus or not is a fact about configurations (or configuration-histories), not about wave functions over configurations.
Yes, some wave functions assign more amplitude to configurations in which the photon passes through the apparatus than others. Still, this does not allow discrete knowledge of the wave function to follow from discrete knowledge of measurements.
The Bohm interpretation, on the other hand, has an answer to this question. When we know a fact, we know a fact about the true configuration-history, which is an element of the theory.
In a sense, the Bohm interpretation states that indexical information about which world we are in is part of fundamental reality, unlike the many-worlds interpretation which states that fundamental reality contains no indexical information. (I have discussed the trouble of indexicals with respect to physicalism previously)
Including such indexical information as "part of reality" means that discrete knowledge is possible, as the discrete knowledge is knowledge of this indexical information.
For this reason, I significantly prefer the Bohm interpretation over the many-worlds interpretation, while acknowledging that there is a great deal of uncertainty here and that there may be a much better interpretation possible. Though my reservations about the many-worlds interpretation had led me to be ambivalent about the comparison between the many-worlds interpretation and the Copenhagen interpretation, I am not similarly ambivalent about Bohm versus many-worlds; I significantly prefer the Bohm interpretation to both many-worlds and to the Copenhagen interpretation.