"Both were interested in how information theory could help illuminate quantum mechanics... the information you gain when you learn something about a system is mathematically defined to be the reduction in your uncertainty about it."
No. This is a fundamental misunderstanding about the nature of information, that has utterly confused physicists for generations.
"But the instant you make that measurement, the wavefunction collapses into one possibility or another"
No. There is no physical wavefunction - it is merely a computational tool.
Quantum theory only describes the probability of detecting a particle. It does not describe actual measurements, of anything, at all. Think about it. In the famous double slit experiment, the only thing ever observed/detected are spots on a screen, or detection counts of particles. The position of the particle being detected, is NEVER actually measured. Rather, which detector (from a set of detectors) detected the particle, is first observed, then the position of that detector, not the detected particle, is measured, and then, the particle's position is INFERRED, not measured, as being the same as the detector.
Nothing else is even a possibility, since the Heisenberg uncertainty principle is equivalent to the statement that only a single bit of information is being manifested, in a quantum detection process: exactly enough information to answer one yes/no question "Was something just detected?", and nothing more.
"Making sense of this measurement problem is the 'most fundamental problem in all of quantum mechanics'..." Exactly. To make sense of it, you have to first recognize the fact, that no "measurement" is actually being performed on the detected entity - only a single-bit detection-decision (AKA wavefunction collapse) is ever being performed.
Rob McEachern