In chapter 1 of Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang, I came across this paragraph on quantum teleportation,
Intuitively, things look pretty bad for Alice. She doesn’t know the state $\lvert\psi\rangle$ of the qubit she has to send to Bob, and the laws of quantum mechanics prevent her from determining the state when she only has a single copy of $\lvert\psi\rangle$ in her possession. What’s worse, even if she did know the state $\lvert\psi\rangle$, describing it precisely takes an infinite amount of classical information since $\lvert\psi\rangle$ takes values in a continuous space. So even if she did know $\lvert\psi\rangle$, it would take forever for Alice to describe the state to Bob.
So Alice and Bob share a qubit each from an EPR pair created long ago and now Alice wishes to teleport the state $\lvert\psi\rangle$ to Bob by only sending classical information.
I do not understand why describing $\lvert\psi\rangle$ takes an infinite amount of classical information, since to my knowledge, only the amplitudes of the basis vectors need to be known($\lvert\psi\rangle=\alpha \lvert 0\rangle+\beta \lvert 1\rangle$). Maybe I did not understand properly what it means for Alice to know a state $\lvert\psi\rangle$. Any guidance would be helpful. Thank you.
PS: I'm not from a Quantum mechanics background.