Looking at the wave function changes nothing. It's all the same physics, and all the same concepts. You can't get different behavior from the physical system by "looking at the wave function" instead of looking at the particles.

Alice hands Bob a particle that is either spin up or a superposition of spin up and spin down. Bob can not distinguish those two cases with a measurement. But if Bob would know the wave function he could tell the difference and Bob could learn the wave function be repeating the experiment many times. So there is a difference between looking at the outcome of a measurement and looking at the wave function.

... via real particles. Which can't go faster than the speed of light.

If Alice measures her particle of an entangled pair the wave function of Bobs particle will change without any particle transporting any information from Alice to Bob. Bob is unable to detect that change with a local measurement but the wave function changed and no particle has been exchanged.

That... doesn't make any sense from a physics point of view. You're trying to understand which is good. But you're mangling the concepts.

What is wrong with that? The wave function fully describes the state of a system but you can not learn that state in general because you would have to perform several different measurements and those measurements do not commute in general and change the state to an eigenstate of the measured observable. Only in an experimental setup with repeated measurements on identically prepared systems can you learn the wave function of the system.

No. You're learning the probability distribution of the wave function. i.e the statistical distribution of the wave function, taken over many measurements.

No, I said identically prepared systems, so the wave functions are identical in every run of the experiment. I then obtain a distribution of the eigenstates of the operator by repeating the experiment and that determines the wave function up to the phase.

I'm not even sure what that means. It's based on a misunderstanding of the underlying concepts, so the sentence doesn't really make sense to me.

Alice prepares a set of entangled particle pairs. On two thirds of the sets she measures her particle and sorts them by outcome. Now Alice has three sets, unmeasured, measured up and measured down. Bob can determine which set is which by measuring his particles, he gets 50/50 up and down, 100 % down and 100 % up respectively.

I never said that.

You did or at least I understood it that way. You say that information is what an observer can learn about the state of a system by performing measurements but there is also the information describing the state of the system, the wave function, which is not necessarily accessible to an observer.

The wave function is a probability distribution.

It is not, it is a description of the state of the system. You can obtain a probability distribution of eigenstates by repeatedly performing measurements on identical wave functions and that distribution is determined by the magnitude of the amplitude of the wave function but the wave function itself is not a probability distribution.

You can learn new information through measurements... but not when the underlying measurements are random. i.e. with entangled particles, all you measure is that (in this case) half of the cards are red, and half are blue. Which you already knew.

Since you already knew that half of the cards are red and half are blue, the measurements give you no new information.

I did not dispute that, observers obtain information about the state of a system by performing measurements, but that is not the same as the information describing the state of the system. The later is the truth, the former is what an observer knows about the truth.