Information is transmitted when you go from a state of "missing knowledge" to a state where you "have knowledge". The word information theorists use for this entropy - entropy signifies how much missing knowledge you have and information is gained whenever you reduce entropy.
Consider the following situation. You send me a signal and I know that your signal will decode to either 0 or 1. I don't know which outcome I will get but I know it would be one of those two outcomes. Before I decode your signal, assuming you send 0 or 1 with equal probability, I was in a state of zero knowledge about the value of the bit. This corresponds to the maximum entropy of $H = 1$. After I decode it, I gained full information about the bit and reduced my entropy to zero.
Similarly, consider a signal where if I measure it, I would obtain 0, 1, 2, or 3. 783
When I actually measure this signal, I've gained two bits of information.
The idea in quantum mechanics is that you can choose any real value of $\alpha$ when you make a qubit in the state $\sqrt{\alpha}\vert 0\rangle + \sqrt{1-\alpha}\vert 1\rangle$. Moreover, the receipient could measure in any basis they like. You might think that the sender could therefore choose $\alpha$ from a set of, say, ${0, 0.1, 0.2... 1}$ (or indeed even more precise) and send the qubit and the recipient could measure the qubit and extract this value of $\alpha$. That's a lot of information sent in just one qubit. However, <PERSON>'s theorem states that this doesn't work - in the end, you can only really send 1 bit of information in 1 qubit.
Note that your examples where you posit a way to transmit infinite information in a signal are not connected to the argument being made by <PERSON> and <PERSON>. Even if you have finite but good precision on $\alpha$, you still can only send 1 bit using a qubit. As for whether analog signals can be used to transmit infinite information, that is addressed by the answer linked in the comment to your question. 855