Let's back up.

You're using colloquial English to reason about physics. This is wrong. Physicists have very specific definitions of the terms they use. Which are sort of similar to the common ones, but differ in key points. Those key points are what you're getting hung up on. And it's difficult to explain the differences without explaining all pf physics.

On top of that, you've disagreed with the common definitions used by physicists "I'm not sure that's correct". Well, it is. If you think that's wrong, you either think you're smarter than all physicists alive, or you're wrong. Pick one.

> ... But if Bob would know the wave function he could tell the difference ...

No. No. A thousand times no.

It just doesn't work like that.

> 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 is a bunch of pseudo-physics words put together in a sentence.

>> You're trying to understand which is good. But you're mangling the concepts.
> What is wrong with that?

Everything. If you're not using the correct terms in the correct way, you might as well be putting random words together in a sentence.

>> 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.

So... you know better than the nuclear physicist.

This should set off alarm bells that you either don't understand the topic, or you don't care to understand it.

> 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.

<sigh> The observer can do multiple measurements to determine the wave function.

>> The wave function is a probability distribution.
> It is not, it is a description of the state of the system.

Again, you're arguing with the nuclear physicist.

The probability distribution is a description of the system.

>> 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,

Yes.

> 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.

<sigh> I'm trying to educate you on physics, and you're arguing pseudo-philosophical metaphysics.

Please stop. Your understanding of the terms is largely wrong. As a result, your arguments are based on falsities, and therefore also wrong.

Please go read a popular book about QM before discussing this with anyone.

And stop arguing with the nuclear physicist. This is one situation where I can appeal to authority without it being a logical fallacy. I've explained repeatedly why you're wrong. You don't seem to understand.